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I recently uploaded a paper to the arXiv entitled Knots with small rational genus, joint with Cameron Gordon. The genesis of this paper was a couple of nice (and related) talks at Caltech by Matthew Hedden and Jake Rasmussen in 2007. They both talked about potential applications of the theory of knot Floer homology to the Berge conjecture. A Berge knot is a (tame) knot $K$ in the 3-sphere which lies on a genus two Heegaard surface, and with the property that on each side of the Heegaard surface there is a meridian disk that the knot intersects exactly once. Equivalently, the inclusion of the knot into each (closed) handlebody sends the generator of $\pi_1(K)$ to a generator of $\pi_1(\text{handlebody})$. Note that since the 3-sphere admits a unique (up to isotopy) Heegaard splitting of any genus, one may think of such a knot as lying on a specific genus 2 surface in $S^3$. Such knots were classified by Berge; they admit (Dehn) surgeries which result in (nontrivial) Lens spaces. The Berge conjecture is the converse; i.e.:

Berge Conjecture: Let $K$ be a knot in $S^3$ which admits a nontrivial Lens space surgery; i.e. there is a Lens space $L$ and a knot $K'$ in $L$ for which $S^3 - K$ is homeomorphic to $L - K'$. Then $K$ is a Berge knot.

An equivalent formulation (of course) is to try to classify knots in Lens spaces which admit an $S^3$ surgery, i.e. to identify the knots $K'$ as in the formulation of the conjecture above. The equivalent formulation says that these knots should be 1-bridge. The strategy of Hedden-Rasmussen (building on work of Ken Baker and Eli Grigsby) to approach the Berge conjecture depends on characterizing such knots by properties which can be detected by topological invariants that behave well under surgery. An example of such a topological invariant is the Casson invariant $\lambda(\cdot)$, a $\mathbb{Z}$-valued invariant of integer homology spheres which satisfies the surgery formula $\lambda(M_{n+1}) - \lambda(M_n) = \text{Arf}(K)$ where $M_i$ denotes the result of $1/i$ surgery on some integral homology sphere $M$ along a fixed knot $K$, and $\text{Arf}(K)$ is the Arf invariant. For more sophisticated invariants like knot Floer homology, the surgery formula is replaced by an exact triangle. One important piece of topological information that is detected by knot Floer homology is the genus of a knot. The approach to the Berge conjecture thus rests on Ken Baker’s impressive paper showing that small genus knots (in a sense to be made precise) in Lens spaces have small bridge number.

Hedden remarked in his talk that his work, and that of his collaborators “gave the first examples of an infinite family of knots that were characterized by their knot Floer homology”. Though technically true, I think this overstates the role of knot Floer homology in this case, since the knots (1-bridge knots in Lens spaces) are entirely characterized (up to isotopy) by their genus (and therefore by any topological invariant which detects genus). My immediate instinct was to think that knots with small genus in any 3-manifold should always be quite special, and that a complete classification might even be feasible. My paper with Cameron confirms this suspicion, and gives such a classification. Let me admit at this point that I am not especially interested in the Berge conjecture per se, although I find it interesting that new ideas in 3-manifold topology are starting to have something meaningful to say about it. In any case, I shall not have anything else to say about it (meaningful or otherwise) in this post.

First I should say that I have been using the word “genus” in a somewhat sloppy manner. For an oriented knot $K$ in $S^3$, a Seifert surface is a compact oriented embedded surface $\Sigma \subset S^3$ whose boundary is $K$. The genus of such a surface is a non-negative integer, and the least such genus over all Seifert surfaces is (said to be) the genus of $K$, denoted $g(K)$. Such a surface represents the generator in the relative homology group $H_2(S^3, K)$ which equals $H_1(K) = \mathbb{Z}$ since $S^3$ has vanishing homology in dimensions 1 and 2. This relative homology group is dual to $H^1(S^3 - K)$, which is parameterized by homotopy classes of maps from $S^3 - K$ to a circle (which is a $K(\mathbb{Z},1)$). The preimage of a regular value under a smooth map dual to the homology class is a smooth proper surface in $S^3 - K$ whose closure is a Seifert surface. It is immediate that $g(K)=0$ if and only if $K$ is an unknot; in other words, the unknot is “characterized” by its genus. There are infinitely many knots of any positive genus in $S^3$; on the other hand, there are only two fibered genus 1 knots — the trefoil and the figure 8 knot (three if you distinguish the left-handed from the right-handed trefoil), and it is worth remarking (from the point of view of the motivation of characterizing knots by topological invariants) that a theorem of Yi Ni says that fiberedness of knots can be detected by knot Floer homology.

For knots in integral homology $3$-spheres, the situation is very similar: every knot admits a Seifert surface, and the least genus of such a surface is the genus of a knot. The unknot is (always) characterized by the fact that it has genus $0$, but there are infinitely many knots of every positive genus. For a knot $K$ in a general $3$-manifold $M$ it is not so easy to define genus. A necessary and sufficient condition for $K$ to bound an embedded surface in its complement is that $[K]=1$ in $H_1(M)$. However, if $[K]$ has finite order, one can find an open properly embedded surface $\Sigma$ in the complement of $K$ whose “boundary” wraps some number of times around $K$. Technically, let $\Sigma$ be a compact oriented surface, and $f:\Sigma \to M$ a map which restricts to an embedding from the interior of $\Sigma$ into $M-K$, and which restricts to an oriented covering map from $\partial \Sigma$ to $K$ (note that we allow $\Sigma$ to have multiple boundary components). If $p$ is the degree of the covering map $\partial \Sigma \to K$, we call $\Sigma$ a $p$-Seifert surface, and define the rational genus of $\Sigma$ to be $-\chi^-(\Sigma)/2p$, where $\chi$ denotes Euler characteristic, and $\chi^-(\Sigma) = \min(0,\chi(\Sigma))$ (for a connected surface $\Sigma$). The reason to use Euler characteristic instead of genus is that Euler characteristic is multiplicative under coverings (unlike genus), and behaves well with respect to “local” operations on surfaces like cut-and-paste. Moreover, (negative) Euler characteristic, unlike genus, is a good measure of complexity for surfaces with possibly many boundary components. The coefficient of $2$ in the denominator reflects the fact that genus is “almost” $-2$ times Euler characteristic. With this definition, we say that the rational genus of $K$, for any knot $K \subset M$ with $[K]$ of finite order in $H_1(M)$, is the infimum of $-\chi^-(\Sigma)/2p$ over all $p$-Seifert surfaces for $K$ and all $p$. The purpose of our paper is to give a complete classification of knots with sufficiently small rational genus, and to show that such knots are always “geometric” — i.e. they can be isotoped into a normal form which is sensitive to the geometric decomposition of the ambient $3$-manifold $M$. Thus the concept of rational genus makes contact between the homological world of the Thurston norm, knot Floer homology and such invariants, and the geometric world of hyperbolic structures, JSJ decompositions and so on.

It is worth pointing out at this point that knots with small rational genus are not special by virtue of being rare: if $K$ is any knot in $S^3$ (for instance) of genus $g(K)$, and $K'$ in $M$ is obtained by $p/q$ Dehn surgery on $K$, then the knot $K'$ has order $p$ in $H_1(M)$, and $\|K'\| \le (g-1/2)/2p$. Since for “most” coprime $p/q$ the integer $p$ is arbitrarily large, it follows that “most” knots obtained in this way have arbitrarily small rational genus.

There is a precise connection between rational genus and the Thurston norm. There is an exact sequence in homology, which contains the fragment $H_2(M,K) \to H_1(K) \to H_1(M)$. Since $H_1(K) = \mathbb{Z}$, the kernel of $H_1(K) \to H_1(M)$ is generated by some class $n[K]$, and one can define the affine subspace $\partial^{-1}(n[K]) \subset H_2(M,K)$. By excision, we identify $H_2(M,K)$ with $H_2(M-\text{int}(N(K)), \partial N(K))$ where $N(K)$ is a tubular neighborhood of $K$. Under this identification, the rational genus of $K$ is equal to $\inf \|[\Sigma]\|_T/2$ where $\|\cdot\|_T$ denotes the (relative) Thurston norm, and the infimum is taken over classes in $H_2(M-\text{int}(N(K)), \partial N(K))$ in the affine subspace corresponding to $\partial^{-1}(n[K])$. Since the Thurston norm is a convex piecewise rational function, this infimum is realized at some rational point. In other words, rational genus of any knot is rational, and is realized by some $p$-Seifert surface, where $n$ as above divides $p$ (note: if $M$ is a rational homology sphere, then necessarily $p=n$, but if the rank of $H_1(M)$ is positive, this is not necessarily true, and $p/n$ might be arbitrarily large). This relationship to the Thurston norm also gives a straightforward algorithm to compute rational genus, since one can compute Thurston norm e.g. by linear programming in normal surface space relative to any triangulation.

The precise statement of results depends on the geometric decomposition of the ambient manifold $M$. By the geometrization theorem (of Perelman), a closed, orientable $3$-manifold is either reducible (i.e. contains an embedded sphere that does not bound a ball), or is a Lens space, or is hyperbolic, or is a small Seifert fiber space, or is toroidal (i.e. contains an essential ($\pi_1$-injective) embedded torus). For the record, the complete “classification” is as follows:

Reducible Theorem: Let ${K}$ be a knot in a reducible manifold ${M}$. Then either

1. ${\|K\| \ge 1/12}$; or
2. there is a decomposition ${M = M' \# M''}$, ${K \subset M'}$ and either
1. ${M'}$ is irreducible, or
2. ${(M',K) = (\mathbb{RP}^3,\mathbb{RP}^1)\#(\mathbb{RP}^3,\mathbb{RP}^1)}$

Lens Theorem: Let ${K}$ be a knot in a lens space ${M}$. Then either

1. ${\|K\| \ge 1/24}$; or
2. ${K}$ lies on a Heegaard torus in ${M}$; or
3. ${M}$ is of the form ${L(4k,2k-1)}$ and ${K}$ lies on a Klein bottle in ${M}$ as a non-separating orientation-preserving curve.

Hyperbolic Theorem: Let ${K}$ be a knot in a closed hyperbolic ${3}$-manifold ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is isotopic to a cable of the core of a Margulis tube.

Small SFS Theorem: Let ${M}$ be an atoroidal Seifert fiber space over ${S^2}$ with three exceptional fibers and let ${K}$ be a knot in ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is a cable of an exceptional Seifert fiber of ${M}$; or
4. ${M}$ is a prism manifold and ${K}$ is a fiber in the Seifert fiber structure of ${M}$ over ${\mathbb{RP}^2}$ with at most one exceptional fiber.

Toroidal Theorem: Let ${M}$ be a closed, irreducible, toroidal 3-manifold, and let ${K}$ be a knot in ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is contained in a hyperbolic piece ${N}$ of the JSJ decomposition of ${M}$ and is isotopic either to a cable of a core of a Margulis tube or into a component of ${\partial N}$; or
4. ${K}$ is contained in a Seifert fiber piece ${N}$ of the JSJ decomposition of ${M}$ and either
1. ${K}$ is isotopic to an ordinary fiber or a cable of an exceptional fiber or into ${\partial N}$, or
2. ${N}$ contains a copy ${Q}$ of the twisted ${S^1}$ bundle over the Möbius band and ${K}$ is contained in ${Q}$ as a fiber in this bundle structure;
5. or

6. ${M}$ is a ${T^2}$-bundle over ${S^1}$ with Anosov monodromy and ${K}$ is contained in a fiber.

The constant $1/402$ is presumably not optimal, but reflects the coarseness of certain geometric estimates at a particular step in the argument. Broadly speaking, there are two cases to consider: when the knot complement $M-K$ is hyperbolic, and when it is not. The complement $M-K$ is hyperbolic unless it contains an essential subsurface of non-negative Euler characteristic.

The case that $M-K$ is hyperbolic is conceptually easiest to analyze. Let $\Sigma$ be a surface, embedded in $M$ and with boundary wrapping some number of times around $K$, realizing the rational genus of $K$. The complete hyperbolic structure on $M-K$ may be deformed, adding back $K$ as a cone geodesic. Just as a cone can be obtained from a wedge of paper by gluing the two edges together, the geometry of a cone geodesic is locally modeled on the quotient space obtained from a (3-dimensional hyperbolic) wedge by gluing the two flat faces together. The thinner the wedge, the smaller the cone angle along the geodesic. For all sufficiently small angles $\theta > 0$, Thurston proved that there exists a unique hyperbolic metric on $M$ which is singular along a cone geodesic, isotopic to $K$, with cone angle $\theta$. Call this metric space $M_\theta$. The cone angle can be increased, deforming the geometry in a family of spaces, until one of the following three things happens:

1. The cone angle is increased all the way to $2\pi$, resulting in the complete hyperbolic structure on $M$, in which $K$ is isotopic to an embedded geodesic; or
2. The volume of the family of manifolds $M_\theta$ goes to zero (and either converges after rescaling to a Euclidean cone manifold, or converges after rescaling to have fixed diameter and injectivity radius going to zero everywhere); or
3. The cone locus bumps into itself (this can only happen for $\theta > \pi$).

As the cone angle along $K$ increases, so does the length of the cone geodesic. Simultaneously, the diameter of an embedded tube about this diameter decreases. While the diameter of the tube is big, the deformation can continue. Hodgson-Kerckhoff analyzed the kinds of degenerations that can occur, and obtained universal geometric control on how fast the tube diameter can shrink, or the length of the cone geodesic grow. They showed that the cone angle can be increased (giving rise to a family of singular hyperbolic structures $M_\theta$) either until $\theta = 2\pi$, or until the product $\theta \cdot \ell$, where $\ell$ is the length of the cone geodesic, is at least $1.019675$, at which point the diameter of an embedded tube about this cone geodesic is at least $0.531$. Since $\theta < 2\pi$ in the latter case, one obtains a lower bound on both the length of the cone geodesic and the diameter of an embedded tube, independent of $K$ or $M$.

Now, one would like to use this big tube to conclude that $\|K\|$ is large. This is accomplished as follows. Geometrically, one constructs a $1$-form $\alpha$ which agrees with the length form on the cone geodesic, which is supported in the tube, and which satisfies $\|d\alpha\|\le C$ pointwise for some (universal) constant $C$. Then one uses this $1$-form to control the topology of $\Sigma$. By Stokes theorem, for any surface $S$ homotopic to $\Sigma$ in $M-K$ one has an estimate

$1.019675/2\pi \le \ell = \int_K \alpha = \frac {1}{p} \int_S d\alpha \le \frac {C}{p} \text{area}(S)$

In particular, the area of $S$ divided by $p$ can’t be too small. However, it turns out that one can find a surface $S$ as above with $\text{area}(S) \le -2\pi\chi(S)$; such an estimate is enough to obtain a universal lower bound on $\|K\|$. Such a surface $S$ can be constructed either by the shrinkwrapping method of Calegari-Gabai, or the (related) PL-wrapping method of Soma. Roughly speaking, one uses the cone geodesic as an “obstacle”, and finds a surface $S$ of least area homotopic to $\Sigma$ (rel. boundary) subject to the constraint that it cannot cross the geodesic. Away from the cone geodesic, $S$ looks like an ordinary minimal surface. In particular, its intrinsic curvature is no more than the extrinsic curvature of hyperbolic space, which is $-1$ everywhere. Along the geodesic, $S$ looks like a bedsheet hanging on a clothesline; in particular, it does not accumulate any corners or atoms of positive curvature along this singularity, so the Gauss-Bonnet theorem gives the desired bound on $\text{area}(S)$.

This leaves the case that $M-K$ is not hyperbolic to analyze. As remarked above, this only occurs when $M-K$ contains an essential surface (which might be closed or proper) of non-negative Euler characteristic, i.e. a sphere, a disk, an annulus or a torus. In this case, one tries to make the intersection of $\Sigma$ with this essential surface as simple as possible; if one arranges this just right, every intersection contributes a definite amount to the topology of $\Sigma$, and one can conclude either that $\Sigma$ is complicated (in which case $\|K\|$ is large), or that the intersection is simple, and therefore draw some topological conclusion.

To actually do this in practice is quite complicated, but fortunately it relies on (largely combinatorial) methods developed at length by Gabai, Scharlemann, Gordon and others over the last 30 years to analyze (so-called) “exceptional surgeries”. Of course, the argument is still complicated, and this analysis takes up most of the length of the paper. It is also worth pointing out that every case provided for by the classification above actually occurs, with examples of arbitrarily small rational genus.

This paper raises several natural questions, the most obvious of which is whether the explicit (but quite small) constants can be improved in any way. The constant $1/402$ in the statement of the Toroidal Theorem is really only there to take care of a knot sitting inside a hyperbolic piece in the decomposition; a knot that interacts in a meaningful way with an essential torus necessarily has rational genus at least $1/24$ (for a precise statement, see the paper). As remarked above, knots of (ordinary) genus $1$ are very plentiful, even in $S^3$, and do not “see” any of the ambient geometry, so the wildest and most optimistic guess might be that there is a chance of classifying knots of rational genus at most $1/4$. There are some (very weak) reasons to think that this fraction is critical, at least in some cases, not least of which is the papers of Hedden and Ni mentioned above. But in the hyperbolic case, it is probably not easy to get a better estimate using purely geometric arguments.

Another approach might be to try to substitute another conclusion (again in the hyperbolic case) than that $K$ be isotopic to the cable of a core of a Margulis tube. For instance, one might ask for $K$ to admit an insulator family (of the kind Gabai used here), or one might merely ask that $K$ be unknotted in the universal cover, or satisfy some other condition. This goes to the heart of a very, very difficult and important question, namely how to identify geometric features of codimension 2 objects in (especially hyperbolic) geometric 3-manifolds from purely topological properties. If I am optimistic, then I can imagine that this paper makes a contribution, however small, to this ongoing project.

Last week, Michael Brandenbursky from the Technion gave a talk at Caltech on an interesting connection between knot theory and quasimorphisms. Michael’s paper on this subject may be obtained from the arXiv. Recall that given a group $G$, a quasimorphism is a function $\phi:G \to \mathbb{R}$ for which there is some least real number $D(\phi) \ge 0$ (called the defect) such that for all pairs of elements $g,h \in G$ there is an inequality $|\phi(gh) - \phi(g) - \phi(h)| \le D(\phi)$. Bounded functions are quasimorphisms, although in an uninteresting way, so one is usually only interested in quasimorphisms up to the equivalence relation that $\phi \sim \psi$ if the difference $|\phi - \psi|$ is bounded. It turns out that each equivalence class of quasimorphism contains a unique representative which has the extra property that $\phi(g^n) = n\phi(g)$ for all $g\in G$ and $n \in \mathbb{Z}$. Such quasimorphisms are said to be homogeneous. Any quasimorphism may be homogenized by defining $\overline{\phi}(g) = \lim_{n \to \infty} \phi(g^n)/n$ (see e.g. this post for more about quasimorphisms, and their relation to stable commutator length).

Many groups that do not admit many homomorphisms to $\mathbb{R}$ nevertheless admit rich families of homogeneous quasimorphisms. For example, groups that act weakly properly discontinuously on word-hyperbolic spaces admit infinite dimensional families of homogeneous quasimorphisms; see e.g. Bestvina-Fujiwara. This includes hyperbolic groups, but also mapping class groups and braid groups, which act on the complex of curves.

Michael discussed another source of quasimorphisms on braid groups, those coming from knot theory. Let $I$ be a knot invariant. Then one can extend $I$ to an invariant of pure braids on $n$ strands by $I(\alpha) = I(\widehat{\alpha \Delta})$ where $\Delta = \sigma_1 \cdots \sigma_{n-1}$, and the “hat” denotes plat closure. It is an interesting question to ask: under what conditions on $I$ is the resulting function on braid groups a quasimorphism?

In the abstract, such a question is probably very hard to answer, so one should narrow the question by concentrating on knot invariants of a certain kind. Since one wants the resulting invariants to have some relation to the algebraic structure of braid groups, it is natural to look for functions which factor through certain algebraic structures on knots; Michael was interested in certain homomorphisms from the knot concordance group to $\mathbb{R}$. We briefly describe this group, and a natural class of homomorphisms.

Two oriented knots $K_1,K_2$ in the $3$-sphere are said to be concordant if there is a (locally flat) properly embedded annulus $A$ in $S^3 \times [0,1]$ with $A \cap S^3 \times 0 = K_1$ and $A \cap S^3 \times 1 = K_2$. Concordance is an equivalence relation, and the equivalence classes form a group, with connect sum as the group operation, and orientation-reversed mirror image as inverse. The only subtle aspect of this is the existence of inverses, which we briefly explain. Let $K$ be an arbitrary knot, and let $K^!$ denote the mirror image of $K$ with the opposite orientation. Arrange $K \cup K^!$ in space so that they are symmetric with respect to reflection in a dividing plane. There is an immersed annulus $A$ in $S^3$ which connects each point on $K$ to its mirror image on $K^!$, and the self-intersections of this annulus are all disjoint embedded arcs, corresponding to the crossings of $K$ in the projection to the mirror. This annulus is an example of what is called a ribbon surface. Connect summing $K$ to $K^!$ by pushing out a finger of each into an arc in the mirror connects the ribbon annulus to a ribbon disk spanning $K \# K^!$. A ribbon surface (in particular, a ribbon disk) can be pushed into a (smoothly) embedded surface in a $4$-ball bounding $S^3$. Puncturing the $4$-ball at some point on this smooth surface, one obtains a concordance from $K\#K^!$ to the unknot, as claimed.

The resulting group is known as the concordance group $\mathcal{C}$ of knots. Since connect sum is commutative, this group is abelian. Notice as above that a slice knot — i.e. a knot bounding a locally flat disk in the $4$-ball — is concordant to the unknot. Ribbon knots (those bounding ribbon disks) are smoothly slice, and therefore slice, and therefore concordant to the trivial knot. Concordance makes sense for codimension two knots in any dimension. In higher even dimensions, knots are always slice, and in higher odd dimensions, Levine found an algebraic description of the concordance groups in terms of (Witt) equivalence classes of linking pairings on a Seifert surface; (some of) this information is contained in the signature of a knot.

Let $K$ be a knot (in $S^3$ for simplicity) with Seifert surface $\Sigma$ of genus $g$. If $\alpha,\beta$ are loops in $\Sigma$, define $f(\alpha,\beta)$ to be the linking number of $\alpha$ with $\beta^+$, which is obtained from $\beta$ by pushing it to the positive side of $\Sigma$. The function $f$ is a bilinear form on $H_1(\Sigma)$, and after choosing generators, it can be expressed in terms of a matrix $V$ (called the Seifert matrix of $K$). The signature of $K$, denoted $\sigma(K)$, is the signature (in the usual sense) of the symmetric matrix $V + V^T$. Changing the orientation of a knot does not affect the signature, whereas taking mirror image multiplies it by $-1$. Moreover, if $\Sigma_1,\Sigma_2$ are Seifert surfaces for $K_1,K_2$, one can form a Seifert surface $\Sigma$ for $K_1 \# K_2$ for which there is some sphere $S^2 \in S^3$ that intersects $\Sigma$ in a separating arc, so that the pieces on either side of the sphere are isotopic to the $\Sigma_i$, and therefore the Seifert matrix of $K_1 \# K_2$ can be chosen to be block diagonal, with one block for each of the Seifert matrices of the $K_i$; it follows that $\sigma(K_1 \# K_2) = \sigma(K_1) + \sigma(K_2)$. In fact it turns out that $\sigma$ is a homomorphism from $\mathcal{C}$ to $\mathbb{Z}$; equivalently (by the arguments above), it is zero on knots which are topologically slice. To see this, suppose $K$ bounds a locally flat disk $\Delta$ in the $4$-ball. The union $\Sigma':=\Sigma \cup \Delta$ is an embedded bicollared surface in the $4$-ball, which bounds a $3$-dimensional Seifert “surface” $W$ whose interior may be taken to be disjoint from $S^3$. Now, it is a well-known fact that for any oriented $3$-manifold $W$, the inclusion $\partial W \to W$ induces a map $H_1(\partial W) \to H_1(W)$ whose kernel is Lagrangian (with respect to the usual symplectic pairing on $H_1$ of an oriented surface). Geometrically, this means we can find a basis for the homology of $\Sigma'$ (which is equal to the homology of $\Sigma$) for which half of the basis elements bound $2$-chains in $W$. Let $W^+$ be obtained by pushing off $W$ in the positive direction. Then chains in $W$ and chains in $W^+$ are disjoint (since $W$ and $W^+$ are disjoint) and therefore the Seifert matrix $V$ of $K$ has a block form for which the lower right $g \times g$ block is identically zero. It follows that $V+V^T$ also has a zero $g\times g$ lower right block, and therefore its signature is zero.

The Seifert matrix (and therefore the signature), like the Alexander polynomial, is sensitive to the structure of the first homology of the universal abelian cover of $S^3 - K$; equivalently, to the structure of the maximal metabelian quotient of $\pi_1(S^3 - K)$. More sophisticated “twisted” and $L^2$ signatures can be obtained by studying further derived subgroups of $\pi_1(S^3 - K)$ as modules over group rings of certain solvable groups with torsion-free abelian factors (the so-called poly-torsion-free-abelian groups). This was accomplished by Cochran-Orr-Teichner, who used these methods to construct infinitely many new concordance invariants.

The end result of this discussion is the existence of many, many interesting homomorphisms from the knot concordance group to the reals, and by plat closure, many interesting invariants of braids. The connection with quasimorphisms is the following:

Theorem(Brandenbursky): A homomorphism $I:\mathcal{C} \to \mathbb{R}$ gives rise to a quasimorphism on braid groups if there is a constant $C$ so that $|I([K])| \le C\cdot\|K\|_g$, where $\|\cdot\|_g$ denotes $4$-ball genus.

The proof is roughly the following: given pure braids $\alpha,\beta$ one forms the knots $\widehat{\alpha\Delta}$, $\widehat{\beta\Delta}$ and $\widehat{\alpha\beta\Delta}$. It is shown that the connect sum $L:= \widehat{\alpha \Delta} \# \widehat{\beta\Delta} \# \widehat{\alpha\beta\Delta}^!$ bounds a Seifert surface whose genus may be universally bounded in terms of the number of strands in the braid group. Pushing this Seifert surface into the $4$-ball, the hypothesis of the theorem says that $I$ is uniformly bounded on $L$. Properties of $I$ then give an estimate for the defect; qed.

It would be interesting to connect these observations up to other “natural” chiral, homogeneous invariants on mapping class groups. For example, associated to a braid or mapping class $\phi \in \text{MCG}(S)$ one can (usually) form a hyperbolic $3$-manifold $M_\phi$ which fibers over the circle, with fiber $S$ and monodromy $\phi$. The $\eta$-invariant of $M_\phi$ is the signature defect $\eta(M_\phi) = \int_Y p_1/3 - \text{sign}(Y)$ where $Y$ is a $4$-manifold with $\partial Y = M_\phi$ with a product metric near the boundary, and $p_1$ is the first Pontriagin form on $Y$ (expressed in terms of the curvature of the metric). Is $\eta$ a quasimorphism on some subgroup of $\text{MCG}(S)$ (eg on a subgroup consisting entirely of pseudo-Anosov elements)?

If $f$ is a smooth function on a manifold $M$, and $p$ is a critical point of $f$, recall that the Hessian $H_pf$ is the quadratic form $\nabla df$ on $T_pM$ (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of $f$ at $p$). Since $H_pf$ is symmetric, it has a well-defined index, which is the dimension of the subspace of maximal dimension on which $H_pf$ is negative definite. The Hessian does not depend on a choice of metric. One way to see this is to give an alternate definition $H_pf(X(p),Y(p)) = X(Yf)(p)$ where $X$ and $Y$ are any two vector fields with given values $X(p)$ and $Y(p)$ in $T_pM$. To see that this does not depend on the choice of $X,Y$, observe

$X(Yf)(p) - Y(Xf)(p) = [X,Y]f(p) = df([X,Y])_p = 0$

because of the hypothesis that $df$ vanishes at $p$. This calculation shows that the formula is symmetric in $X$ and $Y$. Furthermore, since $X(Yf)(p)$ only depends on the value of $X$ at $p$, the symmetry shows that the result only depends on $X(p)$ and $Y(p)$ as claimed. A critical point is nondegenerate if $H_pf$ is nondegenerate as a quadratic form.

In Morse theory, one uses a nondegenerate smooth function $f$ (i.e. one with isolated nondegenerate critical points), also called a Morse function, to understand the topology of $M$: the manifold $M$ has a (smooth) handle decomposition with one $i$-handle for each critical point of $f$ of index $i$. In particular, nontrivial homology of $M$ forces any such function $f$ to have critical points (and one can estimate their number of each index from the homology of $M$). Morse in fact applied his construction not to finite dimensional manifolds, but to the infinite dimensional manifold of smooth loops in some finite dimensional manifold, with arc length as a “Morse” function. Critical “points” of this function are closed geodesics. Any closed manifold has a nontrivial homotopy group in some dimension; this gives rise to nontrivial homology in the loop space. Consequently one obtains the theorem of Lyusternik and Fet:

Theorem: Let $M$ be a closed Riemannian manifold. Then $M$ admits at least one closed geodesic.

In higher dimensions, one can study the space of smooth maps from a fixed manifold $S$ to a Riemannian manifold $M$ equipped with various functionals (which might depend on extra data, such as a metric or conformal structure on $S$). One context with many known applications is when $M$ is a Riemannian $3$-manifold, $S$ is a surface, and one studies the area function on the space of smooth maps from $S$ to $M$ (usually in a fixed homotopy class). Critical points of the area function are called minimal surfaces; the name is in some ways misleading: they are not necessarily even local minima of the area function. That depends on the index of the Hessian of the area function at such a point.

Let $\rho(t)$ be a (compactly supported) $1$-parameter family of surfaces in a Riemannian $3$-manifold $M$, for which $\rho(0)$ is smoothly immersed. For small $t$ the surfaces $\rho(t)$ are transverse to the exponentiated normal bundle of $\rho(0)$; hence locally we can assume that $\rho$ takes the form $\rho(t,u,v)$ where $u,v$ are local co-ordinates on $\rho(0)$, and $\rho(\cdot,u,v)$ is contained in the normal geodesic to $\rho(0)$ through the point $\rho(0,u,v)$; we call such a family of surfaces a normal variation of surfaces. For such a variation, one has the following:

Theorem (first variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0) = f\nu$ where $\nu$ is the unit normal vector field to $\rho(0)$. Then there is a formula:

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,\mu\rangle d\text{area}$

where $\mu$ is the mean curvature vector field along $\rho(0)$.

Proof: let $T,U,V$ denote the image under $d\rho$ of the vector fields $\partial_t,\partial_u,\partial_v$. Choose co-ordinates so that $u,v$ are conformal parameters on $\rho(0)$; this means that $\langle U,V\rangle = 0$ and $\|U\|=\|V\|$ at $t=0$.

The infinitesimal area form on $\rho(t)$ is $\sqrt{\|U\|^2\|V\|^2 - \langle U,V \rangle^2} dUdV$ which we abbreviate by $E^{1/2}$, and write

$\frac d {dt} \text{area}(\rho(t)) = \int_{\rho(t)} \frac {dUdV} {2E^{1/2}} (\|U\|^2\langle V,V\rangle' + \|V\|\langle U,U\rangle' - 2\langle U,V\rangle\langle U,V\rangle')$

Since $V,T$ are the pushforward of coordinate vector fields, they commute; hence $[V,T]=0$, so $\nabla_T V = \nabla_V T$ and therefore

$\langle V,V\rangle' = 2\langle \nabla_T V,V\rangle = 2\langle \nabla_V T,V\rangle = 2(V\langle T,V\rangle - \langle T,\nabla_V V\rangle)$

and similarly for $\langle U,U\rangle'$. At $t = 0$ we have $\langle T,V\rangle = 0$, $\langle U,V\rangle = 0$ and $\|U\|^2 = \|V\|^2 = E^{1/2}$ so the calculation reduces to

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle T,\nabla_U U + \nabla_V V\rangle dUdV$

Now, $T|_{t=0} = f\nu$, and $\nabla_U U + \nabla_V V = \mu E^{1/2}$ so the conclusion follows. qed.

As a corollary, one deduces that a surface is a critical point for area under all smooth compactly supported variations if and only if the mean curvature $\mu$ vanishes identically; such a surface is called minimal.

The second variation formula follows by a similar (though more involved) calculation. The statement is:

Theorem (second variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0)=f\nu$. Suppose $\rho(0)$ is minimal. Then there is a formula:

$\frac {d^2} {dt^2} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,L(f)\nu\rangle d\text{area}$

where $L$ is the Jacobi operator (also called the stability operator), given by the formula

$L = \text{Ric}(\nu) + |A|^2 + \Delta_\rho$

where $A$ is the second fundamental form, and $\Delta_\rho = -\nabla^*\nabla$ is the metric Laplacian on $\rho(0)$.

This formula is frankly a bit fiddly to derive (one derivation, with only a few typos, can be found in my Foliations book; a better derivation can be found in the book of Colding-Minicozzi) but it is easy to deduce some significant consequences directly from this formula. The metric Laplacian on a compact surface is negative self-adjoint (being of the form $-X^*X$ for some operator $X$), and $L$ is obtained from it by adding a $0$th order perturbation, the scalar field $|A|^2 + \text{Ric}(\nu)$. Consequently the biggest eigenspace for $L$ is $1$-dimensional, and the eigenvector of largest eigenvalue cannot change sign. Moreover, the spectrum of $L$ is discrete (counted with multiplicity), and therefore the index of $-L$ (thought of as the “Hessian” of the area functional at the critical point $\rho(0)$) is finite.

A surface is said to be stable if the index vanishes. Integrating by parts, one obtains the so-called stability inequality for a stable minimal surface $S$:

$\int_S (\text{Ric}(\nu) + |A|^2)f^2d\text{area} \le \int_S |\nabla f|^2 d\text{area}$

for any reasonable compactly supported function $f$. If $S$ is closed, we can take $f=1$. Consequently if the Ricci curvature of $M$ is positive, $M$ admits no stable minimal surfaces at all. In fact, in the case of a surface in a $3$-manifold, the expression $\text{Ric}(\nu) + |A|^2$ is equal to $R - K + |A|^2/2$ where $K$ is the intrinsic curvature of $S$, and $R$ is the scalar curvature on $M$. If $S$ has positive genus, the integral of $-K$ is non-negative, by Gauss-Bonnet. Consequently, one obtains the following theorem of Schoen-Yau:

Corollary (Schoen-Yau): Let $M$ be a Riemannian $3$-manifold with positive scalar curvature. Then $M$ admits no immersed stable minimal surfaces at all.

On the other hand, one knows that every $\pi_1$-injective map $S \to M$ to a $3$-manifold is homotopic to a stable minimal surface. Consequently one deduces that when $M$ is a $3$-manifold with positive scalar curvature, then $\pi_1(M)$ does not contain a surface subgroup. In fact, the hypothesis that $S \to M$ be $\pi_1$-injective is excessive: if $S \to M$ is merely incompressible, meaning that no essential simple loop in $S$ has a null-homotopic image in $M$, then the map is homotopic to a stable minimal surface. The simple loop conjecture says that a map $S \to M$ from a $2$-sided surface to a $3$-manifold is incompressible in this sense if and only if it is $\pi_1$-injective; but this conjecture is not yet known.

Update 8/26: It is probably worth making a few more remarks about the stability operator.

The first remark is that the three terms $\text{Ric}(\nu)$, $|A|^2$ and $\Delta$ in $L$ have natural geometric interpretations, which give a “heuristic” justification for the second variation formula, which if nothing else, gives a handy way to remember the terms. We describe the meaning of these terms, one by one.

1. Suppose $f \equiv 1$, i.e. consider a variation by flowing points at unit speed in the direction of the normals. In directions in which the surface curves “up”, the normal flow is focussing; in directions in which it curves “down”, the normal flow is expanding. The net first order effect is given by $\langle \nu,\mu\rangle$, the mean curvature in the direction of the flow. For a minimal surface, $\mu = 0$, and only the second order effect remains, which is $|A|^2$ (remember that $A$ is the second fundamental form, which measures the infinitesimal deviation of $S$ from flatness in $M$; the mean curvature is the trace of $A$, which is first order. The norm $|A|^2$ is second order).
2. There is also an effect coming from the ambient geometry of $M$. The second order rate at which a parallel family of normals $\nu$ along a geodesic $\gamma$ diverge is $\langle R(\gamma',\nu)\gamma',\nu\rangle$ where $R$ is the curvature operator. Taking the average over all geodesics $\gamma$ tangent to $S$ at a point gives the Ricci curvature in the direction of $\nu$, i.e. $\text{Ric}(\nu)$. This is the infinitesimal expansion of area of a geodesic plane under the normal flow, and has second order. The interactions between these terms have higher order, so the net contribution when $f \equiv 1$ is $\text{Ric}(\nu) + |A|^2$.
3. Finally, there is the contribution coming from $f$ itself. Imagine that $S$ is a flat plane in Euclidean space, and let $S_\epsilon$ be the graph of $\epsilon f$. The infinitesimal area element on $S_\epsilon$ is $\sqrt{1+\epsilon^2 |\nabla f|^2} \sim 1+\epsilon^2/2 |\nabla f|^2$. If $f$ has compact support, then differentiating twice by $\epsilon$, and integrating by parts, one sees that the (leading) second order term is $\Delta f$. When $S$ is not totally geodesic, and the ambient manifold is not Euclidean space, there is an interaction which has higher order; the leading terms add, and one is left with $L = \text{Ric}(\nu) + |A|^2 + \Delta$.

The second remark to make is that if the support of a variation $f$ is sufficiently small, then necessarily $|\nabla f|$ will be large compared to $f$, and therefore $-L$ will be positive definite. In other words all variations of a (fixed) minimal surface with sufficiently small support are area increasing — i.e. a minimal surface is locally area minimizing (this is local in the surface itself, not in the “space of all surfaces”). This is a generalization of the important fact that a geodesic in a Riemannian manifold is locally length minimizing (though typically not globally length minimizing).

One final remark is that when $|A|^2$ is big enough at some point $p \in S$, and when the injectivity radius of $S$ at $p$ is big enough (depending on bounds on $\text{Ric}(\nu)$ in some neighborhood of  $p$), one can find a variation with support concentrated near $p$ that violates the stability inequality. Contrapositively, as observed by Schoen, knowing that a minimal surface in a $3$-manifold $M$ is stable gives one a priori control on the size of $|A|^2$, depending only on the Ricci curvature of $M$, and the injectivity radius of the surface at the point. Since stability is preserved under passing to covers (for $2$-sided surfaces, by the fact that the largest eigenvalue of $L$ can’t change sign!) one only needs a lower bound on the distance from $p$ to $\partial S$. In particular, if $S$ is a closed stable minimal surface, there is an a priori pointwise bound on $|A|^2$. This fact has many important topological applications in $3$-manifold topology. On the other hand, when $S$ has boundary, the curvature can be arbitrarily large. The following example is due to Thurston (also see here for a discussion):

Example (Thurston): Let $\Delta$ be an ideal simplex in $\mathbb{H}^3$ with ideal simplex parameter imaginary and very large. The four vertices of $\Delta$ come in two pairs which are very close together (as seen from the center of gravity of the simplex); let $P$ be an ideal quadrilateral whose edges join a point in one pair to a point in the other. The simplex $\Delta$ is bisected by a “square” of arbitrarily small area; together with four “cusps” (again, of arbitrarily small area) one makes a (topological) disk spanning $P$ with area as small as desired. Isotoping this disk rel. boundary to a least area (and therefore stable) representative can only decrease the area further. By the Gauss-Bonnet formula, the curvature of such a disk must get arbitrarily large (and negative) at some point in the interior.

Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety.

Editorial note: I have latexified Jeremy’s email; hence “dhat-mu” becomes $\hat{d}\mu$, “boundary-hat” becomes $\hat{d}$, and “boundary-tilde” becomes $\tilde{d}$. I also linkified the link to Caroline Series’ paper.

Hi Danny,

I was busy with the conference on Thursday and Friday, and taking a break on Saturday, and now I’ve finally had a chance to read your blog, and reply to your message. I decided (especially as Jesse had requested it) to write out a complete outline of the theorem. I’m sending a copy of this message to you, Jesse Johnson, Ian Agol, and Francois Labourie: you are all welcome to reproduce it, as long as it is reproduced in its entirety, and states clearly that this is joint work with Vladimir Markovic. Of course, time and energy permitting, I’ll be happy to answer any questions.

Here is an outline of the argument, working backwards to make it clearer:

1. We want to construct a surface made out of skew pants, each of which has complex half-length close to $R$, and which are joined together so that the complex twist-bends are within $o(1/R)$ of $1$. Using a paper of Caroline
Series (published in the Pacific J. of Mathematics) we show that these surfaces are quasi-isometrically embedded in the universal cover of the three-manifold.

2. Consider the following two conditions on two Borel measures $\mu$ and $\nu$ on a metric space $X$ with the same (finite) total measure:

A. For every Borel subset $A$ of $X$, $\mu(A)$ is less than or equal to the $\nu$-measure of an $\epsilon$ neighborhood of $A$.

B. There is a measure space $(Y, \eta)$ and functions $f: Y \to X$ and $g: Y \to X$ such that $\mu$ and $\nu$ are the push-forwards by $f$ and $g$ respectively of the measure $\eta$, and the distance in $X$ between $f(y)$ and $g(y)$ is less than $\epsilon$ for almost every $y \in Y$.

It is easy to show that B implies A (also that A is symmetric in $\mu$ and $\nu$!). In the case where $\mu$ and $\nu$ are discrete and integral measures (the measure of every point is a non-negative integer), we can show that A implies B (and $Y$ will be a finite set with the counting measure) using Hall’s marriage theorem. In fact, the statement that A implies B for discrete and integral measures is easily shown to be equivalent to Hall’s marriage theorem. I don’t know if A implies B in general because I don’t know how to replace the inductive algorithm for Hall’s marriage theorem with a method that works for a relation between two general measure spaces.

We call $\mu$ and $\nu$ $\epsilon$-equivalent if they satisfy condition A, and note that the condition is additively transitive: if $\mu$ is $\epsilon$-equivalent to $\nu$, and $\nu$ is $\delta$-equivalent to $\rho$, then $\mu$ and $\rho$ are $(\epsilon+\delta)$-equivalent.

3. Suppose that $\gamma$ is one boundary component of a pair of skew pants $P$. We can form the common orthogonals in $P$ from $\gamma$ to each of other other two cuffs. For each common orthogonal, at the point where it meets $\gamma$, we can find a unit normal vector to $\gamma$ that points along this common orthogonal. The two resulting normal vectors are related by a translation along the half-length of $\gamma$ (the suitable square root of the loxodromic element for $\gamma$), so we will call them a pair of opposite unit normal vectors (or pounv for short) and they live in the live in the bundle of pounv’s which is conformally equivalent to the complex plane mod the lattice generated by the half-length of $\gamma$ and $2\pi i$. We give the bundle of pounv’s the Euclidean metric inherited from the complex plane, and also the Lebesgue measure.

4. Given a measure on pants we can produce a measure on the union pounv bundles of the boundary geodesics as follows: if the measure is a unit atom on one pair of skew pants, the resulting measure on pounv bundles is a unit atom on the pounv bundle of each the cuffs, at the pounv described in step 3. We extend to a general measure by linearity. This produces a linear operator we will call the $\hat{d}$ operator.

If we are given a positive integral formal sum of pants (or a multi-set of pants) we can think of it as an integral measure on the space of pants.

5. On the pounv bundle for each closed geodesic we can apply a translation of $1 + i \pi$; we will call this translation $\tau$. We can think of $\tau$ as a map from the union of the pounv bundles to itself.

6. Let $\mu$ be an integral measure on pants with cuff half-lengths close to $R$. We can apply the $\hat{d}$ operator described in step 4 to obtain a measure on the union of pounv bundles of all the boundary geodesics; we will call the measure $\hat{d}\mu$. If $\hat{d}\mu$ and the translation of $\hat{d}\mu$ by $\tau$ are $\epsilon/R$ equivalent, then we can take two oriented pants for each pair of pants in our multi-set (taking each of the two possible orientations) and then fit all of these oriented pants into an oriented surface of the type described in step 1. We use Hall’s marriage theorem as described in step 2, and a very small amount of combinatorics.

If the measure $\hat{d}\mu$, restricted to a given pounv bundle, is $\epsilon/R$ equivalent to a rescaling of Lebesgue measure on that torus, then $\hat{d}\mu$ and $\tau$ of $\hat{d}\mu$ are $2\epsilon/R$-equivalent, which is what we wanted.

******************

This is as far as I got in the first talk at Utah, so it would be best to stop and take a breath for a moment. We haven’t really done anything, but we’ve reformulated the problem: the type of surface we want has been well-defined, and the problem of finding this surface has been reformulated as finding a measure on pairs of pants that satisfies a given criterion.

*****************

7. A two-frame for $M$ will comprise a tangent vector and a normal vector both at the same point, unit length and orthogonal. Given a two-frame we can rotate the tangent vector 120 degrees around the normal vector, using the right-hand rule; the orbit of this action is an ordered triple of two-frames, which will call a tripod. We can also rotate 120 degrees in the opposite direction, and obtain an anti-tripod.

8. A connected pair of two-frames is a pair of two frames along with a geodesic segment connecting them. Given $\epsilon$ and $r$, with $r$ large in terms of $\epsilon$, we can find a weighting function on connected two-frames such that the following properties hold whenever the weight is non-zero:

A. The length of the connecting segment is within $\epsilon$ of $r$.

B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then $\epsilon$ with the normal vector of the other two-frame.

C. The angle between the the tangent vector of the two frame and (the tangent vector to) the connecting geodesic segment is exponentially small in $r$.

Moreover,

D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in $r$) to 1.

E. The weighting is geometrically natural, in that it depends only the length of the connecting segment, the angle between the parallel translated normal vectors, and the angles between the connecting segment and the tangent vectors.

We will describe the (relatively simple) weighting function in the end; we will use the exponential mixing of geodesic flow to obtain property D.

9. Given a tripod and an anti-tripod, we can form three pairs of two-frames by pairing the frames in order, and then we can measures (or weightings) on the connected pairs of two-frames, and then form the product measure (or weighting) by multiplying the weights of the three connections. This gives us a weighting on “connected pairs of tripods” (really a tripod and an anti-tripod) that is supported on connections that satisfy properties A, B, and C.

10. We call a perfect connection between two two-frames a geodesic segment that has a length of $r$, and angle of zero between the segment and the tangent vectors, and translates one normal vector to the other. If a tripod and an anti-tripod were connected by three perfect connection, then they would be a 1-dimensional retract of a flat pair of pants with three cuffs of equal length $R$, where $R$ is approximately $r + \log \cos \pi/6$ when $r$ is large. If the tripod and anti-tripod are connected by arcs that satisfy properties A and B, then the connected pair of tripods is still a retract of a skew pair of pants, whose cuffs have half-length within $\epsilon$ (or $10\epsilon$) of $R$. Thus there is a map from good connected pairs of tripods to good pairs of pants, which we will denote by $\pi$.

11. We can let $\tilde{\mu}$ be the measure on connected pairs of tripods, given by integrating the weighting of steps 8 and 9 with respect to the Liouville measure on pairs of tripods (or pairs of two-frames). We then push this measure forward by $\pi$ to obtain a measure $\mu$ on pairs of pants; after finding a rational approximation and clearing denominators, it will be the $\mu$ that was asked for in step 6. We will show that $\hat{d}\mu$ (taking the original irrational $\mu$) is $\epsilon/R$-equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the proof.

12. A partially connected pair of tripods $T$ is a pair of tripods where we have connected two out of the three pairs of two-frames. To a partially connected pair of tripods we can assign a single closed geodesic $\gamma$ that is homotopic to the concatenation (at both ends) of the two connecting segments. If we connect the third pair of two-frames and apply $\pi$ we obtain a pair of pants $P$, and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of $P$ (as described in step 3). We will describe a method for predicting the pounv for $\gamma$ and $P$ knowing only the partially connected tripod $T$: First, lift $T$ to the solid torus cover of $M$ determined by $\gamma$, and then follow geodesic segments from the tangent vectors of the two unconnected two frames of (the lift of) $T$ to the ideal boundary of this $\gamma$-cover. We can connect these two points in the boundary by two geodesics, each of which goes about half-way around this solid torus cover. We can then find the common orthogonals from each of these geodesics to (the lift of) $\gamma$, and then obtain two normal vectors to $\gamma$ pointing along these common orthogonals; it is easy to verify that these are half-way along $\gamma$ from each other (in the complex sense) and hence form a pounv. Property C of the connections between two-frames (and hence tripods) implies that this predicted pounv will be exponentially close (in $r$) to the actually pounv of any pair of pants $P$.

To summarize: given a good connected pair of tripods, we get a good pair of pants $P$, and taking one cuff gamma of $P$, we get a pounv for $\gamma$ as described in step 3. But we only need two out of the three connecting segments to get $\gamma$, and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for $\gamma$ and $P$ to very high accuracy.

13. We can then define the $\tilde{d}$ operator from measures on partially connected pairs of tripods to measures on the pounv bundles for the associated geodesics; this operator is just the linear extension of the operation in step 12. Given a connected pair of tripods, we can get three partially connected pairs of tripods in the obvious way; we can thereby extend $\tilde{d}$ to map measures on connected pairs of tripods to measures on the bundles of pounv’s; because the predicted pounv described in step 12 is exponentially close to the actual pounv described in step 3, the two measures $\tilde{d} \tilde{\mu}$ and $\hat{d}\mu$ are $\exp(-\alpha r)$-equivalent, by the B => A of step 2.

14. For each closed geodesic $\gamma$, we can lift all the partially connected tripods that give $\gamma$ to the $\gamma$ cover of $M$ described in step 12. There is a natural torus action on the normal bundle of $\gamma$, and this extends to an action on all of the solid torus cover associated to $\gamma$. Moreover, it acts on the (lifts of) partially connected tripods, and it does not change the weightings of the two established connecting segments, because of property E of the weighting function.

This is the crucial point: the effective weighting on a partially connected pair of tripods is not just the product of the weights of the two established connections, but that product times the sum of the weights of all possible third connections. By property D of the weighting function, this sum, while not constant, is exponentially close to being constant, so the effective weighting is exponentially close to being invariant under the torus action. Because the predicted pounv for a partially connected pair of tripods is equivariant for the torus action, the measure $\tilde{d} \tilde{\mu}$ is exponentially close to a torus invariant measure on the pounv bundle (which is necessary a rescaling of Lebesgue measure), in the sense that the Radon-Nikodym derivative is exponentially close to 1. It is then an easy lemma that the two measures are exponentially close in the sense of step 2. And then we’re finished: $\hat{d}\mu$ is exponentially close to $\tilde{d} \tilde{\mu}$, which is exponentially close to a rescaling of Lebesgue measure, which is what we wanted (with
overkill) in step 6.

15. It remains only to define the weighting function described in step 8, which is surprisingly simple: We take some left-invariant metric on $\text{PSL}_2(\bf{C})$, and hence on the two-frame bundle for $M$ and its universal cover. Given a connected pair of two-frames in $M$, we lift to the universal cover, to obtain two two-frames $v$ and $w$. We then flow $v$ and $w$ forward by the frame flow for time $r/4$ to obtain $v'$ and $w'$. We let $V$ be the $\epsilon$ neighborhood of $v'$, and $W$ be the $\epsilon$ neighborhood of $w'$, with the tangent vector of $w'$ replaced by its negation. Then the weighting of the connection is the volume of the intersection of $W$ with the image of $V$ under the frame flow for time $r/2$.

Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have $v$ and $w$ downstairs without any connection, and similarly define $v'$, $w'$, $V$ and $W$, then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs $W$ with the frame flow of $V$. By exponential mixing, this converges at the rate $\exp(-\alpha r)$ to the square of the volume of an $\epsilon$ neighborhood, divided by the volume of $M$.

We can normalize the weights by dividing by this constant.

Jeremy

One obvious comment to make is that the argument is remarkably short, and does not depend on any very delicate or complicated analytic estimates (maybe the argument that the glued up surfaces are quasi-geodesic is the most delicate part). It is fair to say that it defies the conventional wisdom in that respect — I was personally very surprised that the general method could be made to work, especially in light of the failure of Bowen’s program. Kudos to Jeremy and Vlad for their boldness and ingenuity.

Another comment to make is that the matching argument is surprisingly robust and general, and I expect it to have many broader applications. One thing I was confused about in my last post seems to be resolved by Jeremy’s sketch above — if I understand it correctly, one first (almost) pairs continuous measures, and only then approximates them by discrete integral measures (with a little bit of combinatorics at the end). And one really does need exponential mixing rather than just mixing.

Incidentally, apropos the matching argument, there are some interesting and well-known variations where things go haywire. For example, papers by Burago-Kleiner and (Curt) McMullen show that there are examples of separated nets in Euclidean space which are not bilipschitz to a lattice (though, interestingly, Curt shows that they are Holder equivalent). No such examples exist in hyperbolic space, because of — nonamenability and Hall’s marriage theorem! Roughly, when trying to match up points in two nets in hyperbolic space, one doesn’t need to look very far because the number of options grows exponentially. This is one reason why Kahn-Markovic need to control the matchings of their measures carefully, because it must be done on a very small scale (where the exponential growth does not kick in).

I thought I would also mention that in case my previous comments lead one to believe otherwise, exponential mixing of the geodesic flow on a hyperbolic manifold is somewhat delicate. Exponential mixing under a flow $g_t$ on a space $X$ preserving a probability measure $\mu$ means that for all (sufficiently nice) functions $f$ and $h$ on $X$, the correlations $\rho(h,f,t):= \int_X h(x)f(g_tx) d\mu - \int_X h(x) d\mu \int_X f(x) d\mu$ are bounded in absolute value by an expression of the form $C_1e^{-tC_2}$ for suitable constants $C_1,C_2$ (which might depend on the analytic quality of $f$ and $h$). For example, one takes $X$ to be the unit tangent bundle of a hyperbolic manifold, and $g_t$ the geodesic flow (i.e. the flow which pushes vectors along the geodesics they are tangent to, at constant speed). Exponential mixing should be contrasted with the much slower mixing of the horocycle flow on a hyperbolic surface, for which the correlation is bounded by an expression like $C_1(\log t)^{C_2}t^{-1}$. The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle $TM$ splits equivariantly under the flow into three subbundles $E^0, E^s, E^u$ where $E^0$ is $1$-dimensional and tangent to the flow, $E^s$ is contracted uniformly exponentially by the flow, and $E^u$ is expanded uniformly exponentially by the flow. The best one knows for (certain) Anosov flows (by Chernov) is that the flow is stretched exponentially mixing, i.e. with an estimate of the form $C_1e^{-\sqrt{t}C_2}$. One knows exponential mixing for the geodesic flow on variable negative curvature surfaces by Dolgopyat, and on certain locally symmetric spaces, using representation theory. See Pollicott’s lecture notes here for more details. I don’t know if exponential mixing for geodesic flows is known on manifolds of variable negative curvature in high dimensions. Also I’d appreciate it if any reader who knows some ergodic theory can confirm/deny/clarify this paragraph . . .

(Update 8/12): Jeremy tells me that he and Vladimir only need “sufficiently high degree polynomial” mixing, so perhaps there is a decent chance the methods can be extended to variable negative curvature.

(Update 10/29): The paper is now available from the arXiv.

I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF; if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial margin).

In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog) takes the following form:

1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e. “cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective

Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.

First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.

Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third prongs much, having paired the first two pairs).

The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match up.

(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)

This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed) surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.

If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective. However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.

Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)

I recently made the final edits to my paper “Positivity of the universal pairing in 3 dimensions”, written jointly with Mike Freedman and Kevin Walker, to appear in Jour. AMS. This paper is inspired by questions that arise in the theory of unitary TQFT’s. An $n+1$-dimensional TQFT (“topological quantum field theory”) is a functor $Z$ from the category of smooth oriented $n$-manifolds and smooth cobordisms between them, to the category of (usually complex) vector spaces and linear maps, that obeys the (so-called) monoidal axiom $Z(A \coprod B) = Z(A) \otimes Z(B)$. The monoidal axiom implies that $Z(\emptyset)=\mathbb{C}$. Roughly speaking, the functor associates to a “spacelike slice” — i.e. to each $n$-manifold $A$ — the vector space of “quantum states” on $A$ (whatever they are), denoted $Z(A)$. A cobordism stands in for the physical idea of the universe and its quantum state evolving in time. An $n+1$-manifold $W$ bounding $A$ can be thought of as a cobordism from the empty manifold to $A$, so $Z(W)$ is a linear map from $\mathbb{C}$ to $Z(A)$, or equivalently, a vector in $Z(A)$ (the image of $1 \in \mathbb{C}$).

Note that as defined above, a TQFT is sensitive not just to the underlying topology of a manifold, but to its smooth structure. One can define variants of TQFTs by requiring more or less structure on the underlying manifolds and cobordisms. One can also consider “decorated” cobordism categories, such as those whose objects are pairs $(A,K)$ where $A$ is a manifold and $K$ is a submanifold of some fixed codimension (usually $2$) and whose morphisms are pairs of cobordisms $(W,S)$ (e.g.  Wilson loops in a $2+1$-dimensional TQFT).

In realistic physical theories, the space of quantum states is a Hilbert space — i.e. it is equipped with a nondegenerate inner product. In particular, the result of pairing a vector with itself should be positive. One says that a TQFT with this property is unitary. In the TQFT, reversing the orientation of a manifold interchanges a vector space with its dual, and pairing is accomplished by gluing diffeomorphic manifolds with opposite orientations. It is interesting to note that many $3+1$-dimensional TQFTs of interest to mathematicians are not unitary; e.g. Donaldson theory, Heegaard Floer homology, etc. These theories depend on a grading, which prevents attempts to unitarize them. It turns out that there is a good reason why this is true, discussed below.

Definition: For any $n$-manifold $S$, let $\mathcal{M}(S)$ denote the complex vector space spanned by the set of $n+1$-manifolds bounding $S$, up to a diffeomorphism fixed on $S$. There is a pairing on this vector space — the universal pairing — taking values in the complex vector space $\mathcal{M}$ spanned by the set of closed $n+1$-manifolds up to diffeomorphism. If $\sum_i a_iA_i$ and $\sum_j b_jB_j$ are two vectors in $\mathcal{M}(A)$, the pairing of these two vectors is equal to the formal sum $\sum_{ij} a_i\overline{b}_j A_i\overline{B}_j$ where overline is complex conjugation on numbers, and orientation-reversal on manifolds, and $A_i\overline{B}_j$ denotes the closed manifold obtained by gluing ${}A_i$ to $\overline{B}_j$ along $S$.

The point of making this definition is the following. If $v \in \mathcal{M}(S)$ is a vector with the property that $\langle v,v\rangle_S = 0$ (i.e. the result of pairing $v$ with itself is zero), then $Z(v)=0$ for any unitary TQFT $Z$. One says that the universal pairing is positive in $n+1$ dimensions if every nonzero vector $v$ pairs nontrivially with itself.

Example: The Mazur manifold $M$ is a smooth $4$-manifold with boundary $S$. There is an involution $\theta$ of $S$ that does not extend over $M$, so $M,\theta(M)$ denote distinct elements of $\mathcal{M}(S)$. Let $v = M - \theta(M)$, their formal difference. Then the result of pairing $v$ with itself has four terms: $\langle v,v\rangle_S = M\overline{M} - \theta(M)\overline{M} - M\overline{\theta(M)} + \theta(M)\overline{\theta(M)}$. It turns out that all four terms are diffeomorphic to $S^4$, and therefore this formal sum is zero even though $v$ is not zero, and the universal pairing is not positive in dimension $4$.

More generally, it turns out that unitary TQFTs cannot distinguish $s$-cobordant $4$-manifolds, and therefore they are insensitive to essentially all “interesting” smooth $4$-manifold topology! This “explains” why interesting $3+1$-dimensional TQFTs, such as Donaldson theory and Heegaard Floer homology (mentioned above) are necessarily not unitary.

One sees that cancellation arises, and a pairing may fail to be positive, if there are some unusual “coincidences” in the set of terms $A_i\overline{B}_j$ arising in the pairing. One way to ensure that cancellation does not occur is to control the coefficients for the terms appearing in some fixed diffeomorphism type. Observe that the “diagonal” coefficients $a_i\overline{a}_i$ are all positive real numbers, and therefore cancellation can only occur if every manifold appearing as a diagonal term is diffeomorphic to some manifold appearing as an off-diagonal term. The way to ensure that this does not occur is to define some sort of ordering or complexity on terms in such a way that the term of greatest complexity can occur only on the diagonal. This property — diagonal dominance — can be expressed in the following way:

Definition: A pairing $\langle \cdot,\cdot \rangle_S$ as above satisfies the topological Cauchy-Schwarz inequality if there is a complexity function $\mathcal{C}$ defined on all closed $n+1$-manifolds, so that if ${}A,B$ are any two $n+1$-manifolds with boundary $S$, there is an inequality $\mathcal{C}(A\overline{B}) \le \max(\mathcal{C}(A\overline{A}),\mathcal{C}(B\overline{B}))$ with equality if and only if $A=B$.

The existence of such a complexity function ensures diagonal dominance, and therefore the positivity of the pairing $\langle\cdot,\cdot\rangle_S$.

Example: Define a complexity function $\mathcal{C}$ on closed $1$-manifolds, by defining $\mathcal{C}(M)$ to be equal to the number of components of $M$. This complexity function satisfies the topological Cauchy-Schwarz inequality, and proves positivity for the universal pairing in $1$ dimension.

Example: A suitable complexity function can also be found in $2$ dimensions. The first term in the complexity is number of components. The second is a lexicographic list of the Euler characteristics of the resulting pieces (i.e. the complexity favors more components of bigger Euler characteristic). The first term is maximized if and only if the pieces of $A$ and $B$ are all glued up in pairs with the same number of boundary components in $S$; the second term is then maximized if and only if each piece of $A$ is glued to a piece of $B$ with the same Euler characteristic and number of boundary components — i.e. if and only if $A=B$.

Positivity holds in dimensions below $3$, and fails in dimensions above $3$. The main theorem we prove in our paper is that positivity holds in dimension $3$, and we do this by constructing an explicit complexity function which satisfies the topological Cauchy-Schwarz inequality.

Unfortunately, the function itself is extremely complicated. At a first pass, it is a tuple $c=(c_0,c_1,c_2,c_3)$ where $c_0$ treats number of components, $c_1$ treats the kernel of $\pi_1(S) \to \pi_1(A)$ under inclusion, $c_2$ treats the essential $2$-spheres, and $c_3$ treats prime factors arising in the decomposition.

The term $c_1$ is itself very interesting: for each finite group $G$ Witten and Dijkgraaf constructed a real unitary TQFT $Z_G$ (i.e. one for which the resulting vector spaces are real), so that roughly speaking $Z_G(S)$ is the vector space spanned by representations of $\pi_1(S)$ into $G$ up to conjugacy, and $Z_G(A)$ is the vector that counts (in a suitable sense) the number of ways each such representation extends over $\pi_1(A)$. The value of $Z_G$ on a closed manifold is roughly just the number of representations of the fundamental group in $G$, up to conjugacy. The complexity $c_1$ is obtained by first enumerating all isomorphism classes of finite groups $G_1,G_2,G_3 \cdots$ and then listing the values of $Z_{G_i}$ in order. If the kernel of $\pi_1(S) \to \pi_1(A)$ is different from the kernel of $\pi_1(S) \to \pi_1(B)$, this difference can be detected by some finite group (this fact depends on the fact that $3$-manifold groups are residually finite, proved in this context by Hempel); so $c_1$ is diagonal dominant unless these two kernels are equal; equivalently, if the maximal compression bodies of $S$ in $A$ and $B$ are diffeomorphic rel. $S$. It is essential to control these compression bodies before counting essential $2$-spheres, so this term must come before $c_2$ in the complexity.

The term $c_3$ has a contribution $c_p$ from each prime summand. The complexity $c_p$ itself is a tuple $c_p = (c_S,c_h,c_a)$ where $c_S$ treats Seifert-fibered pieces, $c_h$ treats hyperbolic pieces, and $c_a$ treats the way in which these are assembled in the JSJ decomposition. The term $c_h$ is quite interesting; evaluated on a finite volume hyperbolic $3$-manifold $M$ it gives as output the tuple $c_h(M) = (-\text{vol}(M),\sigma(M))$ where $\text{vol}(M)$ denotes hyperbolic volume, and $\sigma(M)$ is the geodesic length spectrum, or at least those terms in the spectrum with zero imaginary part. The choice of the first term depends on the following theorem:

Theorem: Let $S$ be an orientable surface of finite type so that each component has negative Euler characteristic, and let ${}A,B$ be irreducible, atoroidal and acylindrical, with boundary $S$. Then $A\overline{A},A\overline{B},B\overline{B}$ admit unique complete hyperbolic structures, and either $2\text{vol}(A\overline{B}) > \text{vol}(A\overline{A})+\text{vol}(B\overline{B})$ or else $2\text{vol}(A\overline{B}) = \text{vol}(A\overline{A}) + \text{vol}(B\overline{B})$ and $S$ is totally geodesic in $A\overline{B}$.

This theorem is probably the most technically difficult part of the paper. Notice that even though in the end we are only interested in closed manifolds, we must prove this theorem for hyperbolic manifolds with cusps, since these are the pieces that arise in the JSJ decomposition. This theorem was proved for closed manifolds by Agol-Storm-Thurston, and our proof follows their argument in general terms, although there are more technical difficulties in the cusped case. One starts with the hyperbolic manifold $A\overline{B}$, and finds a least area representative of the surface $S$. Cut along this surface, and double (metrically) to get two singular metrics on the topological manifolds $A\overline{A}$ and $B\overline{B}$. The theorem will be proved if we can show the volume of this singular metric is bigger than the volume of the hyperbolic metric. Such comparison theorems for volume are widely studied in geometry; in many circumstances one defines a geometric invariant of a Riemannian metric, and then shows that it is minimized/maximized on a locally symmetric metric (which is usually unique in dimensions $>2$). For example, Besson-Courtois-Gallot famously proved that a negatively curved locally symmetric metric on a manifold uniquely minimizes the volume entropy over all metrics with fixed volume (roughly, the entropy of the geodesic flow, at least when the curvature is negative).

Hamilton proved that if one rescales Ricci flow to have constant volume, then scalar curvature $R$ satisfies $R' = \Delta R + 2|\text{Ric}_0|^2 + \frac 2 3 R(R-r)$ where $\text{Ric}_0$ denotes the traceless Ricci tensor, and $r$ denotes the spatial average of the scalar curvature $R$. If the spatial minimum of $R$ is negative, then at a point achieving the minimum, $\Delta R$ is non-negative, as are the other two terms; in other words, if one does Ricci flow rescaled to have constant volume, the minimum of scalar curvature increases (this fact remains true for noncompact manifolds, if one substitutes infimum for maximum). Conversely, if one rescales to keep the infimum of scalar curvature constant, volume decreases under flow. In $3$ dimensions, Perelman shows that Ricci flow with surgery converges to the hyperbolic metric. Surgery at finite times occurs when scalar curvature blows up to positive infinity, so surgery does not affect the infimum of scalar curvature, and only makes volume smaller (since things are being cut out). Consequently, Perelman’s work implies that of all metrics on a hyperbolic $3$-manifold with the infimum of scalar curvature equal to $-6$, the constant curvature metric is the unique metric minimizing volume.

Now, the metric on $A\overline{A}$ obtained by doubling along a minimal surface is not smooth, so one cannot even define the curvature tensor. However, if one interprets scalar curvature as an “average” of Ricci curvature, and observes that a minimal surface is flat “on average”, then one should expect that the distributional scalar curvature of the metric is equal to what it would be if one doubled along a totally geodesic surface, i.e. identically equal to $-6$. So Perelman’s inequality should apply, and prove the desired volume estimate.

To make this argument rigorous, one must show that the singular metric evolves under Ricci flow, and instantaneously becomes smooth, with $R \ge -6$. A theorem of Miles Simon says that this follows if one can find a smooth background metric with uniform bounds on the curvature and its first derivatives, and which is $1+\epsilon$-bilipschitz to the singular metric. The existence of such a background metric is essentially trivial in the closed case, but becomes much more delicate in the cusped case. Basically, one needs to establish the following comparison lemma, stated somewhat informally:

Lemma: Least area surfaces in cusps of hyperbolic $3$-manifolds become asymptotically flat faster than the thickness of the cusp goes to zero.

In other words, if one lifts a least area surface $S$ to a surface $\tilde{S}$ in the universal cover, there is a (unique) totally geodesic surface $\pi$ (the “osculating plane”) asymptotic to $\tilde{S}$ at the fixed point of the parabolic element corresponding to the cusp, and satisfying the following geometric estimate. If $B_t$ is the horoball centered at the parabolic fixed point at height $t$ (for some horofunction), then the Hausdorff distance between $\tilde{S} \cap B_t$ and $\pi \cap B_t$ is $o(e^{-t})$. One must further prove that if a surface $S$ has multiple ends in a single cusp, these ends osculate distinct geodesic planes. Given this, it is not too hard to construct a suitable background metric. Between ends of $S$, the geometry looks more and more like a slab wedged between two totally geodesic planes. The double of this is a nonsingular hyperbolic manifold, so it certainly enjoys uniform control on the curvature and its first derivatives; this gives the background metric in the thin part. In the thick part, one can convolve the singular metric with a bump function to find a bilipschitz background metric; compactness of the thick part implies trivially that any smooth metric enjoys uniform bounds on the curvature and its first derivatives. Hence one may apply Simon, and then Perelman, and the volume estimate is proved.

The Seifert fibered case is very fiddly, but ultimately does not require many new ideas. The assembly complexity turns out to be surprisingly involved. Essentially, one thinks of the JSJ decomposition as defining a decorated graph, whose vertices correspond to the pieces in the decomposition, and whose edges control the gluing along tori. One must prove an analogue of the topological Cauchy-Schwarz inequality in the context of (decorated) graphs. This ends up looking much more like the familiar TQFT picture of tensor networks, but a more detailed discussion will have to wait for another post.