2009 Pacific Northwest Geometry Seminar

Lagrangian Mean Curvature flow for entire Lipschitz graphs, Albert Chau (University of British Columbia)

Abstract: We prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs.  As an application of the stimates for the solution, we establish a Bernstein type result for translating solitons. The results are from joint work with Jingyi Chen and Weiyong He.

Collapsing Sequences of Constant Mean Curvature Surfaces in Riemannian Manifolds, Adrian Butscher (Stanford)

Abstract: Since every Riemannian manifold is locally Euclidean up to second order, small geodesic spheres of radius r have nearly constant mean curvature of magnitude 2/r.  However, it is known that there are fairly restrictive conditions under which a geodesic sphere of sufficiently small radius can be perturbed to have exactly constant mean curvature.  I will investigate a related question: whether it is possible to assemble small geodesic spheres into extended surfaces of near-constant mean curvature by the gluing technique, and perturbing these surfaces to have exactly constant mean curvature.  It turns out that the conditions under which this is possible are the result of a subtle interplay between the surface and the background geometry.  A result of my investigation is a construction of small constant mean curvature surfaces that locally resemble classical Delaunay surfaces but exhibit new global properties that are impossible in the classical setting.  In addition, I will relate my investigation to the following question: given a sequence \Sigma_r of surfaces of mean curvature 2/r contained within a tubular neighbourhood of size O(r) of a lower-dimensional variety \Gamma in a Riemannian manifold M, what can be said about \Gamma? 

The isoperimetric type inequalities and nonlinear PDEs, Pengfei Guan (McGill)

Abstract: We discuss Alexandrov-Fenchel type isoperimetric
inequalities and their relationship with elliptic and parabolic
type partial differential equations. We give a proof of the
isoperimetric inequality for quermassintegrals of non-convex
starshaped domains, using an expanding geometric curvature flow. We also prove a similar type of inequalities for functions
on $\mathbb S^n$ using elliptic Hessian equation.

The rate of change of width under flows, Bill Minicozzi (Johns Hopkins)

Abstract:  I will discuss a geometric invariant, that we call the width, of a manifold and first show how it can be realized as the sum of areas of minimal 2-spheres.  When $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we will estimate the rate of change of width under various geometric Flows, including flows by mean curvature and Ricci curvature, to prove sharp estimates for extinction times.  This is joint work with Toby Colding.

Smooth Metric Measure Spaces, Guofang Wei (UCSB)

Abstract: Smooth metric measure spaces are Riemannian manifolds with a conformal change of the Riemannian measure and occur naturally as measured Gromov-Hausdorff limit of Riemannian manifolds. The important curvature quantity here is the Bakry-Emery Ricci tensor, which  corresponds to the (synthetic) Ricci curvature lower bound for (nonsmooth) metric measure spaces. What geometric and topological results for Ricci curvature can be extended to Bakry-Emery Ricci tensor? Recently there are many developments. We will discuss comparison geometry and rigidity in this direction. 

A general regularity theory for stable codimension 1 integral varifolds, Neshan Wickramasekera (Cambridge)

Abstract: The focus of this talk will be a new regularity theorem for  the class of singular stable minimal hypersurfaces  (stable codimension 1 integral varifolds) of an open all. Making no assumption a priori on the size of the singular set, the theorem gives a natural, geometric structural condition or a hypersurface in this class to be smooth and embedded in he interior up to a lower dimensional, generally unavoidable, singular  set. Precisely, suppose that a hypersurface in this class has the property that  no singular point has a neighborhood in which the hypersurface is a union of  $C^{1, \alpha}$ (for some arbitrarily chosen  \alpha \in (0, 1)$) hypersurfaces-with-boundary meeting
(only) along their common boundary. Then it is smooth  and embedded away from the boundary of the ball and away from a
possible interior singular set of codimension at least 7  (which is empty if the dimension of the hypersurface is $\leq 6$). The
work generalizes the regularity theory  of R. Schoen and L. Simon. Some applications as well as what can be said  in the absence of the above structural condition will also be discussed.