Analysis and Geometric Analysis Seminar
An isoperimetric profile of a Riemannian manifold is a function that for each positive number $V$ associates the optimal perimeter needed bound a volume equal to $V$. On this talk we'll see how for convex co-compact hyperbolic 3-manifolds this relates to Renormalized Volume (a studied functional on the deformation space). We will use this relation together with some tools from General relativity (namely the Hawking mass) to prove that, in the appropriate setup, the isoperimetric profile of a hyperbolic 3-manifold stays below the profile of a model, and equality occurs if and only if the manifold is isometric to the model. This is joint work with Celso Viana.