Program:

3:10 - 4:00 PM    Pengzi Miao (Univ. of Miami)

4:10 - 5:00 PM    Jonathan Luk (Stanford Univ.)

Title/Abstract:

Pengzi Miao (University of Miami)

Title:  Minimal hypersurfaces and boundary behavior of compact manifolds with
nonnegative scalar curvature

Abstract:
On a compact Riemannian manifold with boundary having positive mean
curvature, a fundamental result of Shi and Tam states that, if the
manifold has nonnegative scalar curvature and if the boundary is
isometric to a strictly convex hypersurface in the Euclidean space,
then the total mean curvature of the boundary is no greater than the
total mean curvature of the corresponding Euclidean hypersurface. In
3-dimension, Shi-Tam's result is known to be equivalent to the
Riemannian positive mass theorem.

In this talk, we will discuss a supplement to Shi-Tam's theorem
by including the effect of minimal hypersurfaces on a chosen boundary
component. More precisely, we consider a compact manifold with
nonnegative scalar curvature, whose boundary consists of two parts,
the outer boundary and the horizon boundary. Here the horizon
boundary is the union of all closed minimal hypersurfaces in the
manifold and the outer boundary is assumed to be a topological
sphere. In a relativistic context, such a manifold represents a body
surrounding apparent horizon of black holes in a time symmetric
initial data set. By assuming the outer boundary is isometric to a
suitable 2-convex hypersurface in a Schwarzschild manifold of
positive mass m, we establish an inequality relating m, the area of
the horizon boundary, and two weighted total mean curvatures of the
outer boundary and the hypersurface in the Schwarzschild manifold. In
3-dimension, our result is equivalent to the Riemannian Penrose
inequality. This is joint work with Siyuan Lu.

Jonathan Luk (Stanford University)

Title: Strong cosmic censorship in spherical symmetry for two-ended
asymptotically flat data

Abstract:
I will present a recent work (joint with Sung-Jin Oh) on the strong
cosmic censorship conjecture for the
Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for
two-ended asymptotically flat data. For this model, it was previously
proved (by M. Dafermos and I. Rodnianski) that a certain formulation
of the strong cosmic censorship conjecture is false, namely, the
maximal globally hyperbolic development of a data set in this class
is extendible as a Lorentzian manifold with a C0 metric. Our main
result is that, nevertheless, a weaker formulation of the conjecture
is true for this model, i.e., for a generic (possibly large) data set
in this class, the maximal globally hyperbolic development is
inextendible as a Lorentzian manifold with a C2 metric.

In this talk, we introduce the Laplacian eigenvalue problem and briefly go over its history.  Then we will present a recent result which gives a sharp lower bound of the fundamental gap for convex domain of spheres motivated by the modulus of continuity approach introduced by Andrews-Clutterbuck.  This is joint work with Lili Wang and Guofang Wei.

We begin by discussing the natural diffusion associated to mean curvature flow and work of Soner and Touzi showing that, in Euclidean space, this stochastic structure allows one to reformulate mean curvature flow as the solution to a type of stochastic target problem. Then we describe work with Ionel Popescu adapting the target problem formulation to Ricci flow on compact surfaces and using the accompanying diffusion to understand the convergence of the normalized Ricci flow. We aim to avoid being overly technical, instead focusing on the ideas underlying the appearance of stochastic objects in the context of curvature flow.

In 2011 J.Streets and G.Tian introduced a family of metric flows over a complex Hermitian manifold. We consider one particular member of this family and prove that if the initial metric has Griffiths positive Chern curvature, then this property is preserved along the flow.  On a manifold with Griffiths non-negative Chern curvature this flow has nice regularization properties, in particular, for any t>0 the zero set of Chern curvature becomes invariant under certain torsion-twisted parallel transport. If time permits, we discuss applications of the results to some uniformization problems.

Given a compact Riemannian manifold with nonnegative Ricci curvature and convex boundary, it is interesting to estimate its size in terms of the volume, the area of its boundary etc.  I will discuss some open problems and present some partial results.

Joint Seminar with Analysis.