Abstract:
Self-expanding solutions of curvature flows evolve by homothetic expansions under the flow. Rotational symmetric examples are constructed by Ecker-Huisken, Angenent-Chopp-Ilmanen, Helmensdorfer et. al for the Mean Curvature Flow, and by Huisken-Ilmanen, Grugan-Lee-Wheeler et. al for the Inverse Mean Curvature Flow. Many known examples are asymptotic to some standard models such as round cylinders and round cones. In this talk, the speaker will talk about rotational rigidity results for self-expanders of both Mean Curvature and Inverse Mean Curvature Flows, proving that certain self-expanders asymptotic to cones or cylinders are necessarily rotational symmetric. These are joint works with Peter McGrath, and with Gregory Drugan and Hojoo Lee.

Program

3:00-3:50pm  APM 7421 

Ovidiu Munteanu (Univ. of Connecticut)  ``Poisson equation on complete manifolds''

Abstract: I will discuss sharp estimates for the Green's function on
complete manifolds and their applications to solving the Poisson
equation. I will mention new sharp results about existence of
solutions and their asymptotic behavior. Some new results
about gradient Ricci solitons will be presented as application.

4:00-4:50pm  APM 2402  

Jacob Bernstein (Johns Hopkins)  ``Surfaces of Low Entropy''

Abstract: Following Colding and Minicozzi, we consider the entropy of
(hyper)-surfaces in Euclidean space.  This is a numerical measure of
the geometric complexity of the surface.  In addition, this quantity
is intimately tied to to the singularity formation of the mean
curvature flow which is a natural geometric heat flow of
submanifolds.  In the talk, I will discuss several results that show
that closed surfaces for which the entropy is small are simple in
various senses.  This is all joint work with L. Wang.

Abstract:
In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R^3 by Colding and Minicozzi.This is a joint work with Jingyi Chen.

A Hamiltonian Stationary submanifold of complex space is a Lagrangian manifold whose volume is stationary under Hamiltonian variations.  We consider gradient graphs $(x,Du(x))$ for a function $u$.    For a smooth $u$, the Euler-Lagrange equation can be expressed as a fourth order nonlinear equation in $u$ that can be locally linearized (using a change of tangent plane) to the bi-Laplace.  The volume can be defined for lower regularity, however, and computing the Euler-Lagrange equation with less assumed regularity gives a "double divergence" equation of second order quantities.   We show several results.  First, there is a $c_n$ so that if the Hessian $D^2u$ is $c_n$-close to a continuous matrix-valued function, then the potential must be smooth.  Previously, Schoen and Wolfson showed that when the potential was $C^{2,\alpha}$, then the potential $u$ must be smooth.    We are also able to show full regularity when the Hessian is bounded within certain ranges.   This allows us to rule out conical solutions with mild singularities.

This is joint work with Jingyi Chen.