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Monthly Archives: August 2009
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
Cubic forms in differential geometry
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 … Continue reading
Surface subgroups – more details from Jeremy Kahn
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 … Continue reading
Posted in 3manifolds, Ergodic Theory, Surfaces
Tagged Kahn, Markovic, pair of pants, surface groups
4 Comments
Surface subgroups in hyperbolic 3manifolds
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 … Continue reading
Posted in 3manifolds, Ergodic Theory, Surfaces
Tagged Bowen, geodesic flow, Hall's marriage theorem, Kahn, LERF, Markovic, pair of pants, surface groups, Waldhausen
10 Comments
Faces of the scl norm ball
I am in Melbourne at the moment, in the middle of giving a lecture series, as part of the 2009 ClayMahler lectures (also see here). Yesterday I gave a lecture with the title “faces of the scl norm ball”, and … Continue reading
Posted in Dynamics, Groups, Surfaces
Tagged Bavard duality, free groups, immersions, maximal representation, quasimorphisms, Rigidity, rotation number, scl, Surfaces, Symplectic geometry
1 Comment