# 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 co-ordinates, 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 3-manifolds, Ergodic Theory, Surfaces | | 4 Comments

## Surface subgroups in hyperbolic 3-manifolds

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 3-manifolds, Ergodic Theory, Surfaces | | 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 Clay-Mahler 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 | | 1 Comment