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If is a smooth function on a manifold , and is a critical point of , recall that the Hessian is the quadratic form on (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of at ). Since is symmetric, it has a well-defined index, which is the dimension of the subspace of maximal dimension on which is negative definite. The Hessian does not depend on a choice of metric. One way to see this is to give an alternate definition where and are any two vector fields with given values and in . To see that this does not depend on the choice of , observe
because of the hypothesis that vanishes at . This calculation shows that the formula is symmetric in and . Furthermore, since only depends on the value of at , the symmetry shows that the result only depends on and as claimed. A critical point is nondegenerate if is nondegenerate as a quadratic form.
In Morse theory, one uses a nondegenerate smooth function (i.e. one with isolated nondegenerate critical points), also called a Morse function, to understand the topology of : the manifold has a (smooth) handle decomposition with one -handle for each critical point of of index . In particular, nontrivial homology of forces any such function to have critical points (and one can estimate their number of each index from the homology of ). 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 be a closed Riemannian manifold. Then admits at least one closed geodesic.
In higher dimensions, one can study the space of smooth maps from a fixed manifold to a Riemannian manifold equipped with various functionals (which might depend on extra data, such as a metric or conformal structure on ). One context with many known applications is when is a Riemannian -manifold, is a surface, and one studies the area function on the space of smooth maps from to (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 be a (compactly supported) -parameter family of surfaces in a Riemannian -manifold , for which is smoothly immersed. For small the surfaces are transverse to the exponentiated normal bundle of ; hence locally we can assume that takes the form where are local co-ordinates on , and is contained in the normal geodesic to through the point ; 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 be a normal variation of surfaces, so that where is the unit normal vector field to . Then there is a formula:
where is the mean curvature vector field along .
Proof: let denote the image under of the vector fields . Choose co-ordinates so that are conformal parameters on ; this means that and at .
The infinitesimal area form on is which we abbreviate by , and write
Since are the pushforward of coordinate vector fields, they commute; hence , so and therefore
and similarly for . At we have , and so the calculation reduces to
Now, , and 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 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 be a normal variation of surfaces, so that . Suppose is minimal. Then there is a formula:
where is the Jacobi operator (also called the stability operator), given by the formula
where is the second fundamental form, and is the metric Laplacian on .
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 for some operator ), and is obtained from it by adding a th order perturbation, the scalar field . Consequently the biggest eigenspace for is -dimensional, and the eigenvector of largest eigenvalue cannot change sign. Moreover, the spectrum of is discrete (counted with multiplicity), and therefore the index of (thought of as the “Hessian” of the area functional at the critical point ) 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 :
for any reasonable compactly supported function . If is closed, we can take . Consequently if the Ricci curvature of is positive, admits no stable minimal surfaces at all. In fact, in the case of a surface in a -manifold, the expression is equal to where is the intrinsic curvature of , and is the scalar curvature on . If has positive genus, the integral of is non-negative, by Gauss-Bonnet. Consequently, one obtains the following theorem of Schoen-Yau:
Corollary (Schoen-Yau): Let be a Riemannian -manifold with positive scalar curvature. Then admits no immersed stable minimal surfaces at all.
On the other hand, one knows that every -injective map to a -manifold is homotopic to a stable minimal surface. Consequently one deduces that when is a -manifold with positive scalar curvature, then does not contain a surface subgroup. In fact, the hypothesis that be -injective is excessive: if is merely incompressible, meaning that no essential simple loop in has a null-homotopic image in , then the map is homotopic to a stable minimal surface. The simple loop conjecture says that a map from a -sided surface to a -manifold is incompressible in this sense if and only if it is -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 , and in 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.
- Suppose , 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 , the mean curvature in the direction of the flow. For a minimal surface, , and only the second order effect remains, which is (remember that is the second fundamental form, which measures the infinitesimal deviation of from flatness in ; the mean curvature is the trace of , which is first order. The norm is second order).
- There is also an effect coming from the ambient geometry of . The second order rate at which a parallel family of normals along a geodesic diverge is where is the curvature operator. Taking the average over all geodesics tangent to at a point gives the Ricci curvature in the direction of , i.e. . 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 is .
- Finally, there is the contribution coming from itself. Imagine that is a flat plane in Euclidean space, and let be the graph of . The infinitesimal area element on is . If has compact support, then differentiating twice by , and integrating by parts, one sees that the (leading) second order term is . When 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 .
The second remark to make is that if the support of a variation is sufficiently small, then necessarily will be large compared to , and therefore 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 is big enough at some point , and when the injectivity radius of at is big enough (depending on bounds on in some neighborhood of ), one can find a variation with support concentrated near that violates the stability inequality. Contrapositively, as observed by Schoen, knowing that a minimal surface in a -manifold is stable gives one a priori control on the size of , depending only on the Ricci curvature of , and the injectivity radius of the surface at the point. Since stability is preserved under passing to covers (for -sided surfaces, by the fact that the largest eigenvalue of can’t change sign!) one only needs a lower bound on the distance from to . In particular, if is a closed stable minimal surface, there is an a priori pointwise bound on . This fact has many important topological applications in -manifold topology. On the other hand, when has boundary, the curvature can be arbitrarily large. The following example is due to Thurston (also see here for a discussion):
Example (Thurston): Let be an ideal simplex in with ideal simplex parameter imaginary and very large. The four vertices of come in two pairs which are very close together (as seen from the center of gravity of the simplex); let be an ideal quadrilateral whose edges join a point in one pair to a point in the other. The simplex is bisected by a “square” of arbitrarily small area; together with four “cusps” (again, of arbitrarily small area) one makes a (topological) disk spanning 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.
Quadratic forms (i.e. homogeneous polynomials of degree two) are fundamental mathematical objects. For the ancient Greeks, quadratic forms manifested in the geometry of conic sections, and in Pythagoras’ theorem. Riemann recognized the importance of studying abstract smooth manifolds equipped with a field of infinitesimal quadratic forms (i.e. a Riemannian metric), giving rise to the theory of Riemannian manifolds. In contrast to more general norms, an inner product on a vector space enjoys a big group of symmetries; thus infinitesimal Riemannian geometry inherits all the richness of the representation theory of orthogonal groups, which organizes the various curvature tensors and Weitzenbock formulae. It is natural that quadratic forms should come up in so many distinct ways in differential geometry: one uses calculus to approximate a smooth object near some point by a linear object, and the “difference” is a second-order term, which can often be interpreted as a quadratic form. For example:
- If is a Riemannian manifold, at any point one can choose an orthonormal frame for , and exponentiate to obtain geodesic normal co-ordinates. In such local co-ordinates, the metric tensor satisfies and . The second order derivatives can be expressed in terms of the Riemann curvature tensor at .
- If is an immersed submanifold of Euclidean space, at every point there is a unique linear subspace that is tangent to at . The second order difference between these two spaces is measured by the second fundamental form of , a quadratic form (with coefficients in the normal bundle) whose eigenvectors are the directions of (extrinsic) principal curvature. If has codimension one, the second fundamental form is easily described in terms of the Gauss map taking each point on to the unique unit normal to at that point, and using the flatness of the ambient Euclidean space to identify the normal spheres at different points with “the” standard sphere. The second fundamental form is then defined by the formula . For higher codimension, one considers Gauss maps with values in an appropriate Grassmannian.
- If is a smooth function on a manifold , a critical point of is a point at which (i.e. at which all the partial derivatives of in some local coordinates vanish). At such a point, one defines the Hessian , which is a quadratic form on , determined by the second partial derivatives of at such a point. If is a Levi-Civita connection on (determined by an Riemannian metric on compatible with the smooth structure) then . The condition that the Levi-Civita connection is torsion-free translates into the fact that the antisymmetric part of is equal to for any -form ; in this context, this means that the antisymmetric part of the Hessian vanishes — i.e. that it is symmetric (and therefore a quadratic form). If is a different connection, then for some -form , and therefore their values at agree, and is well-defined, independent of a choice of metric.
By contrast, cubic forms are less often encountered, either in geometry or in other parts of mathematics; their appearance is often indicative of unusual richness. For example: Lie groups arise as the subgroups of automorphisms of vector spaces preserving certain structure. Orthogonal and symplectic groups are those that preserve certain (symmetric or alternating) quadratic forms. The exceptional Lie group is the group of automorphisms of that preserves a generic (i.e. nondegenerate) alternating -form. One expects to encounter cubic forms most often in flavors of geometry in which the local transformation pseudogroups are bigger than the orthogonal group.
One example is that of -dimensional complex projective geometry. If is a domain in the Riemann sphere, one can think of as a geometric space in at least two natural ways: by considering the local pseudogroup of all holomorphic self-maps between open subsets of the Riemann sphere, restricted to (i.e. all holomorphic functions), or by considering only those holomorphic maps that extend to the entire Riemann sphere (i.e. the projective transformations: ). The difference between these two geometric structures is measured by a third-order term, called the Schwarzian derivative. If is homeomorphic to a disk, then we can think of as the image of the round unit disk under a uniformizing map . At every point there is a unique projective transformation that osculates to to second order at (i.e. has the same value, first derivative, and second derivative as at the point ); the (scaled) third derivative is the Schwarzian of at . In local co-ordinates, . Actually, although the Schwarzian is sensitive to third-order information, it should really be thought of as a quadratic form on the (one-dimensional) complex tangent space to .
Real projective geometry gives rise to similar invariants. Consider an immersed curve in the (real projective) plane. At every point, there is a unique osculating conic, that agrees with the immersed curve to second order. The projective curvature (really a cubic form) measures the third order deviation between these two immersed submanifolds at this point. See e.g. the book by Ovsienko and Tabachnikov for more details.
Another example is the so-called symplectic curvature. Let be a flat symplectic space; this could be ordinary Euclidean space with its standard symplectic form, or a quotient of such a space by a discrete group of translations. A linear subspace of through the origin is a Lagrangian subspace if it has (maximal) dimension , and the restriction of the symplectic form to is identically zero. A smooth submanifold of dimension is Lagrangian if its tangent space at every point is a Lagrangian submanifold. A Lagrangian submanifold of a flat symplectic space inherits a natural cubic form on the tangent space at every point, which can be defined in any of the following equivalent ways:
- If is a symplectic manifold and is a Lagrangian submanifold, then near any point one can find a neighborhood and choose symplectic coordinates so that is symplectomorphic to a neighborhood of some point in . Moreover, every other Lagrangian submanifold sufficiently close (in ) to can be taken in some possibly smaller neighborhood to be of the form , where is a smooth function on (well-defined up to a constant), thought of as a section of . In the context above, choose local symplectic coordinates (by a linear symplectic transformation) for which the flat space looks locally like and looks locally like . The condition that and are tangent at the origin means that the -jet of vanishes. The first nonvanishing term are the third partial derivatives of , which can be thought of as the coefficients of a (symmetric) cubic form on .
- If we choose a Euclidean metric on compatible with the flat symplectic structure, the second fundamental form of at some point is a quadratic form on with coefficients in the normal bundle to . The symplectic form identifies the normal to with the dual , so by contracting indices, one obtains a cubic form on . This form does not depend on the choice of Euclidean metric, since a different metric skews the normal bundle replacing it with . But since is Lagrangian, the identification of this normal bundle with is insensitive to the skewed term, and therefore independent of the choices.
- The space of all Lagrangian subspaces of is a symmetric space, homeomorphic to , sometimes called the Shilov boundary of the Siegel upper half-space. If and is a tangent vector to in , then one obtains a symmetric quadratic form on in the following way. If is a transverse Lagrangian to , and is a -parameter family of Lagrangians starting at , then for small the Lagrangians and are transverse, and span . For any there is a unique decomposition . Define . Then is a symmetric bilinear form that vanishes on , and therefore descends to a form on that depends only on . A Lagrangian submanifold maps to by the Gauss map . One obtains a cubic form on associated to as follows: if then is a tangent vector to in , and therefore determines a quadratic form on ; this form is then evaluated on the vectors .
One application of symplectic curvature is to homological mirror symmetry, where the symplectic curvature associated to a Lagrangian family of Calabi-Yau -folds in determines the so-called “Yukawa 3-differential”, whose expression in a certain local coordinate gives the generating function for the number of rational curves of degree in a generic quintic hypersurface in . This geometric picture is described explicitly in the work of Givental (e.g. here). In another more recent paper, Givental shows how the topological recursion relations, the string equation and the dilaton equation in Gromov-Witten theory can be reformulated in terms of the geometry of a certain Lagrangian cone in a formal loop space (the geometric property of this cone is that it is overruled — i.e. each tangent space is tangent to the cone exactly along , where is a formal variable). This geometric condition translates into properties of the symplectic curvature of the Lagrangian cone, from which one can read off the “gravitational descendents” in the theory (let me add that this subject is quite far from my area of expertise, and that I come to this material as an interested outsider).
Cubic forms occur naturally in other “special” geometric contexts, e.g. holomorphic symplectic geometry (Rozansky-Witten invariants), affine differential geometry (related to the discussion of the Schwarzian above), etc. Each of these contexts is the start of a long story, which is best kept for another post.
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 , “boundary-hat” becomes , and “boundary-tilde” becomes . I also linkified the link to Caroline Series’ paper.
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 , and which are joined together so that the complex twist-bends are within of . 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 and on a metric space with the same (finite) total measure:
A. For every Borel subset of , is less than or equal to the -measure of an neighborhood of .
B. There is a measure space and functions and such that and are the push-forwards by and respectively of the measure , and the distance in between and is less than for almost every .
It is easy to show that B implies A (also that A is symmetric in and !). In the case where and are discrete and integral measures (the measure of every point is a non-negative integer), we can show that A implies B (and 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 and -equivalent if they satisfy condition A, and note that the condition is additively transitive: if is -equivalent to , and is -equivalent to , then and are -equivalent.
3. Suppose that is one boundary component of a pair of skew pants . We can form the common orthogonals in from to each of other other two cuffs. For each common orthogonal, at the point where it meets , we can find a unit normal vector to that points along this common orthogonal. The two resulting normal vectors are related by a translation along the half-length of (the suitable square root of the loxodromic element for ), 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 and . 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 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 ; we will call this translation . We can think of as a map from the union of the pounv bundles to itself.
6. Let be an integral measure on pants with cuff half-lengths close to . We can apply the 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 . If and the translation of by are 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 , restricted to a given pounv bundle, is equivalent to a rescaling of Lebesgue measure on that torus, then and of are -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 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 and , with large in terms of , 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 of .
B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then 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 .
D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in ) 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 , 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 , where is approximately when 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 (or ) of . Thus there is a map from good connected pairs of tripods to good pairs of pants, which we will denote by .
11. We can let 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 to obtain a measure on pairs of pants; after finding a rational approximation and clearing denominators, it will be the that was asked for in step 6. We will show that (taking the original irrational ) is -equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the proof.
12. A partially connected pair of tripods 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 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 we obtain a pair of pants , and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of (as described in step 3). We will describe a method for predicting the pounv for and knowing only the partially connected tripod : First, lift to the solid torus cover of determined by , and then follow geodesic segments from the tangent vectors of the two unconnected two frames of (the lift of) to the ideal boundary of this -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) , and then obtain two normal vectors to pointing along these common orthogonals; it is easy to verify that these are half-way along 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 ) to the actually pounv of any pair of pants .
To summarize: given a good connected pair of tripods, we get a good pair of pants , and taking one cuff gamma of , we get a pounv for as described in step 3. But we only need two out of the three connecting segments to get , and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for and to very high accuracy.
13. We can then define the 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 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 and are -equivalent, by the B => A of step 2.
14. For each closed geodesic , we can lift all the partially connected tripods that give to the cover of described in step 12. There is a natural torus action on the normal bundle of , and this extends to an action on all of the solid torus cover associated to . 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 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: is exponentially close to , 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 , and hence on the two-frame bundle for and its universal cover. Given a connected pair of two-frames in , we lift to the universal cover, to obtain two two-frames and . We then flow and forward by the frame flow for time to obtain and . We let be the neighborhood of , and be the neighborhood of , with the tangent vector of replaced by its negation. Then the weighting of the connection is the volume of the intersection of with the image of under the frame flow for time .
Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have and downstairs without any connection, and similarly define , , and , then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs with the frame flow of . By exponential mixing, this converges at the rate to the square of the volume of an neighborhood, divided by the volume of .
We can normalize the weights by dividing by this constant.
I will try to add comments as they occur to me.
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 on a space preserving a probability measure means that for all (sufficiently nice) functions and on , the correlations are bounded in absolute value by an expression of the form for suitable constants (which might depend on the analytic quality of and ). For example, one takes to be the unit tangent bundle of a hyperbolic manifold, and 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 . The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle splits equivariantly under the flow into three subbundles where is -dimensional and tangent to the flow, is contracted uniformly exponentially by the flow, and 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 . 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 -manifold contains a closed -injective surface. Equivalently, contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing question in -manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic -manifold groups were LERF, one would be able to deduce that all hyperbolic -manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic -manifold groups are LERF; if successful, this would therefore resolve some of the most important outstanding problems in -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 , 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:
- Given , for a sufficiently big constant , 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 . In fact, one can further insist that the complex length of the boundary geodesic is very close to (i.e. holonomy transport around this geodesic does not rotate the normal bundle very much).
- Conversely, given any geodesic of complex length very close to , 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.
- 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
- One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is -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 is a totally geodesic pair of pants with boundary components of length close to , the pants retract onto a geodesic spine, i.e. an immersed totally geodesic theta graph, whose edges all have length close to , and which meet at angles very close to 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 .
Given a tripod in some plane in the tangent space at some point of , one can exponentiate the edges for length to construct such a half-spine; if and 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 , once 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 , 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 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 -injective, one must use geometry. A closed (immersed) surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is -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 no matter how long its boundary components are. So if the boundary components have length , then at the points where they are thinnest, they are only 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 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 am in Melbourne at the moment, in the middle of giving a lecture series, as part of the 2009 Clay-Mahler lectures (also see here). Yesterday I gave a lecture with the title “faces of the scl norm ball”, and I thought I would try to give a sense of what it was all about. This also gives me an excuse to fiddle around with images in wordpress.
One starts with a basic question: given an immersion of a circle in the plane, when is there an immersion of the disk in the plane that bounds the given immersion of a circle? I.e., given a immersion , when is there an immersion for which factors through ? Obviously this depends on . Consider the following examples:
The first immersed circle obviously bounds an immersed disk; in fact, an embedded disk.
The second circle does not bound such a disk. One way to see this is to use the Gauss map, i.e. the map that takes each point on the circle to the unit tangent to its image under the immersion. The degree of the Gauss map for an embedded circle is (depending on a choice of orientation). If an immersed circle bounds an immersed disk, one can use this immersed disk to define a 1-parameter family of immersions, connecting the initial immersed circle to an embedded immersed circle; hence the degree of the Gauss map is aso for an immersed circle bounding an immersed disk; this rules out the second example.
The third example maps under the Gauss map with degree 1, and yet it does not bound an immersed disk. One must use a slightly more sophisticated invariant to see this. The immersed circle divides the plane up into regions. For each bounded region , let be an embedded arc, transverse to , that starts in the region and ends up “far away” (ideally “at infinity”). The arc determines a homological intersection number that we denote , where each point of intersection contributes depending on orientations. In this example, there are three bounded regions, which get the numbers , , respectively:
If is any map of any oriented surface with one boundary component whose boundary factors through , then the (homological) degree with which maps over each region complementary to the image of is the number we have just defined. Hence if bounds an immersed disk, these numbers must all be positive (or all negative, if we reverse orientation). This rules out the third example.
The complete answer of which immersed circles in the plane bound immersed disks was given by S. Blank, in his Ph.D. thesis at Brandeis in 1967 (unfortunately, this does not appear to be available online). The answer is in the form of an algorithm to decide the question. One such algorithm (not Blank’s, but related to it) is as follows. The image of cuts up the plane into regions , and each region gets an integer . Take “copies” of each region , and think of these as pieces of a jigsaw puzzle. Try to glue them together along their edges so that they fit together nicely along and make a disk with smooth boundary. If you are successful, you have constructed an immersion. If you are not successful (after trying all possible ways of gluing the puzzle pieces together), no such immersion exists. This answer is a bit unsatisfying, since in the first place it does not give any insight into which loops bound and which don’t, and in the second place the algorithm is quite slow and impractial.
As usual, more insight can be gained by generalizing the question. Fix a compact oriented surface and consider an immersed -manifold . One would like to know which such -manifolds bound an immersion of a surface. One piece of subtlety is the fact that there are examples where itself does not bound, but a finite cover of (e.g. two copies of ) does bound. It is also useful to restrict the class of -manifolds that one considers. For the sake of concreteness then, let be a hyperbolic surface with geodesic boundary, and let be an oriented immersed geodesic -manifold in . An immersion is said to virtually bound if the map factors as a composition where the second map is , and where the first map is a covering map with some degree . The fundamental question, then is:
Question: Which immersed geodesic -manifolds in are virtually bounded by an immersed surface?
It turns out that this question is unexpectedly connected to stable commutator length, symplectic rigidity, and several other geometric issues; I hope to explain how in the remainder of this post.
First, recall that if is any group and , the commutator length of , denoted , is the smallest number of commutators in whose product is equal to , and the stable commutator length is the limit . One can geometrize this definition as follows. Let be a space with , and let be a homotopy class of loop representing the conjugacy class of . Then over all surfaces (possibly with multiple boundary components) mapping to whose boundary wraps a total of times around . One can extend this definition to -manifolds in the obvious way, and one gets a definition of stable commutator length for formal sums of elements in which represent in homology. Let denote the vector space of real finite linear combinations of elements in whose sum represents zero in (real group) homology (i.e. in the abelianization of , tensored with ). Let be the subspace spanned by chains of the form and . Then descends to a (pseudo)-norm on the quotient which we denote hereafter by ( for homogeneous).
There is a dual definition of this norm, in terms of quasimorphisms.
Definition: Let be a group. A function is a homogeneous quasimorphism if there is a least non-negative real number (called the defect) so that for all and one has
A function satisfying the second condition but not the first is an (ordinary) quasimorphism. The vector space of quasimorphisms on is denoted , and the vector subspace of homogeneous quasimorphisms is denoted . Given , one can homogenize it, by defining . Then and . A quasimorphism has defect zero if and only if it is a homomorphism (i.e. an element of ) and makes the quotient into a Banach space.
Examples of quasimorphisms include the following:
- Let be a free group on a generating set . Let be a reduced word in and for each reduced word , define to be the number of copies of in . If denotes the corresponding element of , define (note this is well-defined, since each element of a free group has a unique reduced representative). Then define . This quasimorphism is not yet homogeneous, but can be homogenized as above (this example is due to Brooks).
- Let be a closed hyperbolic manifold, and let be a -form. For each let be the geodesic representative in the free homotopy class of . Then define . By Stokes’ theorem, and some basic hyperbolic geometry, is a homogeneous quasimorphism with defect at most .
- Let be an orientation-preserving action of on a circle. The group of homeomorphisms of the circle has a natural central extension , the group of homeomorphisms of that commute with integer translation. The preimage of in this extension is an extension . Given , define ; this descends to a -valued function on , Poincare’s so-called rotation number. But on , this function is a homogeneous quasimorphism, typically with defect .
- Similarly, the group has a universal cover with deck group . The symplectic group acts on the space of Lagrangian subspaces in . This is equal to the coset space , and we can therefore define a function . After picking a basepoint, one obtains an -valued function on the symplectic group, which lifts to a real-valued function on its universal cover. This function is a quasimorphism on the covering group, whose homogenization is sometimes called the symplectic rotation number; see e.g. Barge-Ghys.
Quasimorphisms and stable commutator length are related by Bavard Duality:
Theorem (Bavard duality): Let be a group, and let . Then there is an equality where the supremum is taken over all homogeneous quasimorphisms.
This duality theorem shows that with the defect norm is the dual of with the norm. (this theorem is proved for elements by Bavard, and in generality in my monograph, which is a reference for the content of this post.)
What does this have to do with rigidity (or, for that matter, immersions)? Well, one sees from the examples (and many others) that homogeneous quasimorphisms arise from geometry — specifically, from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). One expects to find rigidity in extremal circumstances, and therefore one wants to understand, for a given chain , the set of extremal quasimorphisms for , i.e. those homogeneous quasimorphisms satisfying . By the duality theorem, the space of such extremal quasimorphisms are a nonempty closed convex cone, dual to the set of hyperplanes in that contain and support the unit ball of the norm. The fewer supporting hyperplanes, the smaller the set of extremal quasimorphisms for , and the more rigid such extremal quasimorphisms will be.
When is a free group, the unit ball in the norm in is a rational polyhedron. Every nonzero chain has a nonzero multiple contained in the boundary of this polyhedron; let denote the face of the polyhedron containing this multiple in its interior. The smaller the codimension of , the smaller the dimension of the cone of extremal quasimorphisms for , and the more rigidity we will see. The best circumstance is when has codimension one, and an extremal quasimorphism for is unique, up to scale, and elements of .
An infinite dimensional polyhedron need not necessarily have any top dimensional faces; thus it is natural to ask: does the unit ball in have any top dimensional faces? and can one say anything about their geometric meaning? We have now done enough to motivate the following, which is the main theorem from my paper “Faces of the scl norm ball”:
Theorem: Let be a free group. For every isomorphism (up to conjugacy) where is a compact oriented surface, there is a well-defined chain . This satisfies the following properties:
- The projective class of intersects the interior of a codimension one face of the norm ball
- The unique extremal quasimorphism dual to (up to scale and elements of ) is the rotation quasimorphism (to be defined below) associated to any complete hyperbolic structure on
- A homologically trivial geodesic -manifold in is virtually bounded by an immersed surface in if and only if the projective class of (thought of as an element of ) intersects . Equivalently, if and only if is extremal for . Equivalently, if and only if .
It remains to give a definition of . In fact, we give two definitions.
First, a hyperbolic structure on and the isomorphism determines a representation . This lifts to , since is free. The composition with rotation number is a homogeneous quasimorphism on , well-defined up to . Note that because the image in is discrete and torsion-free, this quasimorphism is integer valued (and has defect ). This quasimorphism is .
Second, a geodesic -manifold in cuts the surface up into regions . For each such region, let be an arc transverse to , joining to . Let denote the homological (signed) intersection number. Then define .
We now show how 3 follows. Given , we compute as above. Let be such a surface, mapping to . We adjust the map by a homotopy so that it is pleated; i.e. so that is itself a hyperbolic surface, decomposed into ideal triangles, in such a way that the map is a (possibly orientation-reversing) isometry on each ideal triangle. By Gauss-Bonnet, we can calculate . On the other hand, wraps times around (homologically) so where the sign in each case depends on whether the ideal triangle is mapped in with positive or negative orientation. Consequently with equality if and only if the sign of every triangle is . This holds if and only if the map is an immersion; on the other hand, equality holds if and only if is extremal for . This proves part 3 of the theorem above.
Incidentally, this fact gives a fast algorithm to determine whether is the virtual boundary of an immersed surface. Stable commutator length in free groups can be computed in polynomial time in word length; likewise, the value of can be computed in polynomial time (see section 4.2 of my monograph for details). So one can determine whether projectively intersects , and therefore whether it is the virtual boundary of an immersed surface. In fact, these algorithms are quite practical, and run quickly (in a matter of seconds) on words of length 60 and longer in .
One application to rigidity is a new proof of the following theorem:
Corollary (Goldman, Burger-Iozzi-Wienhard): Let be a closed oriented surface of positive genus, and a Zariski dense representation. Let be the Euler class associated to the action. Suppose that (note: by a theorem of Domic and Toledo, one always has ). Then is discrete.
Here is the first Chern class of the bundle associated to . The proof is as follows: cut along an essential loop into two subsurfaces . One obtains homogeneous quasimorphisms on each group (i.e. the symplectic rotation number associated to ), and the hypothesis of the theorem easily implies that they are extremal for . Consequently the symplectic rotation number is equal to , at least on the commutator subgroup. But this latter quasimorphism takes only integral values; it follows that each element in fixes a Lagrangian subspace under . But this implies that is not dense, and since it is Zariski dense, it is discrete. (Notes: there are a couple of details under the rug here, but not many; furthermore, the hypothesis that is Zariski dense is not necessary (but can be derived as a conclusion with more work), and one can just as easily treat representations of compact surface groups as closed ones; finally, Burger-Iozzi-Wienhard prove more than just this statement; for instance, they show that the space of maximal representations is always real semialgebraic, and describe it in some detail).
More abstractly, we have shown that extremal quasimorphisms on are unique. In other words, by prescribing the value of a quasimorphism on a single group element, one determines its values on the entire commutator subgroup. If such a quasimorphism arises from some geometric or dynamical context, this can be interpreted as a kind of rigidity theorem, of which the Corollary above is an example.