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Category Archives: 3manifolds
Filling geodesics and hyperbolic complements
Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flowinvariant way into stable, unstable and flow directions) and contact … Continue reading
Quasigeodesic flows on hyperbolic 3manifolds
My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobiuslike groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress … Continue reading
Rotation numbers and the JankinsNeumann ziggurat
I’m in Melbourne right now, where I recently attended the Hyamfest and the preceding workshop. There were many excellent talks at both the workshop and the conference (more on that in another post), but one thing that I found very interesting … Continue reading
Hyperbolic Geometry Notes #5 – Mostow Rigidity
1. Mostow Rigidity For hyperbolic surfaces, Moduli space is quite large and complicated. However, in three dimensions Moduli space is trivial: Theorem 1 If is a homotopy equivalence of closed hyperbolic manifolds with , then is homotopic to an isometry. … Continue reading
Posted in 3manifolds, Groups, Hyperbolic geometry, Uncategorized
2 Comments
Knots with small rational genus
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 … Continue reading
Quasimorphisms from knot invariants
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 , a quasimorphism … Continue reading
Posted in 3manifolds, Groups
Tagged 4ball genus, braids, CochranOrrTeichner, knot concordance, quasimorphisms, ribbon, signature, slice
2 Comments
Second variation formula for minimal surfaces
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 coordinates, the coefficients of the Hessian are the second partial derivatives of at … Continue reading