## Search

## Joint UCI-UCR-UCSD Southern California Differential Geometry Seminar

**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.