## University of Tennessee - Geometric analysis seminar

### Fall 2017

Thursday, December 7, 17:05

Ryan Unger (UTK)

Title: The Yamabe Problem IV

Abstract: We complete the proof of the Yamabe conjecture in the positive case by giving some details on conformal normal coordinates, the asymptotic expansion of the conformal Green's function, and the test function estimate in the low-dimensional and LCF cases.

Thursday, November 30, 17:05

Alex Mramor (University of California Irvine)

Title: Applications of mean curvature flow with surgery.

Abstract: In this talk I will give a brief introduction to the mean curvature flow with surgery and describe some applications of it to studying the moduli space of hypersurfaces and the level set flow.

Thursday, November 16, 17:05

Brett Kotschwar (Arizona State)

Title: A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons.

Abstract: Shrinking Ricci solitons are generalized fixed points of the Ricci flow equation and models for the geometry of solutions to the flow in the neighborhood of a developing singularity. It is conjectured that every end of a four-dimensional complete noncompact shrinking soliton is smoothly asymptotic to either a cone or a standard cylinder at infinity. I will discuss recent joint work with Lu Wang related to this conjecture in which we prove that a shrinking Ricci soliton which is asymptotic to infinite order along some end to one of the standard cylinders $S^k\times {\mathbb{R}}^{n-k}$ for $k\geq 2$ must actually be isometric to the cylinder on that end.

Thursday, November 9, 17:05

Ryan Unger (UTK)

Title: The Yamabe Problem III

Abstract: We will continue our discussion of the resolution of the Yamabe conjecture in the case when the Yamabe energy is positive.

Thursday, November 2, 17:05

Ryan Unger (UTK)

Title: The Yamabe Problem II

Abstract: In 1960 H. Yamabe conjectured that given a compact Riemannian manifold, there exists a conformal deformation of the metric to one of constant scalar curvature. Here we discuss the resolution of the conjecture in the case when the Yamabe energy is positive. This case has a surprising connection to the positive mass theorem in general relativity.

Thursday, October 26, 17:05

Ryan Unger (UTK)

Title: The Yamabe Problem I

Abstract: In 1960, H. Yamabe conjectured that, given a compact Riemannian manifold, there exists a conformal deformation of the metric to one of constant scalar curvature. The combined works of Yamabe, N. Trudinger, T. Aubin, and R. Schoen gave an affirmative solution in 1984. Here we review Yamabe's original paper and give the proof of Yamabe's conjecture in the case when the Yamabe energy is nonpositive.

Thursday, October 19, 17:05

Kevin Sonnanburg (UTK)

Title: A Liouville Theorem for Ancient Solutions to Mean Curvature Flow

Abstract: A compact surface moving by mean curvature flow eventually develops a singularity. In the study of parabolic blow-ups at singularities, ancient solutions (defined for all negative time) often arise as a result of the time dilation. If we can categorize ancient solutions, we can better understand blow-up limits. One expects such limits to be highly regular with some kind of constant behavior, given the infinite time allowed for diffusion to occur. We give a Liouville-type theorem restricting a certain class of ancient mean curvature flows to spheres or cylinders.

Thursday, October 12, 17:05

Kevin Sonnanburg (UTK)

Title: Blow-up Continuity of Mean-Convex, Type-I Mean Curvature Flow

Abstract: Under mean curvature flow, each point of a hypersurface moves with velocity equal to its mean curvature vector, shrinking the hypersurface's area as rapidly as possible. A closed, embedded hypersurface M(t) shrinks and becomes singular in finite time. One of the most basic questions about a PDE that develops singularities is the relationship between the occurrence of its singularities and its initial data. For certain classes of mean-convex mean curvature flows, we show the first singular time T and the limit set “M(T)” are continuous with respect to the initial hypersurface.

Thursday, September 28, 17:05

Theodora Bourni (UTK)

Title: Ancient pancakes II

Abstract: We show that, up to rigid motions, there is a unique compact, convex, rotationally symmetric, ancient solution of mean curvature flow that lies in a slab of width $\pi$ and in no smaller slab. This is joint work with Mat Langford and Giuseppe Tinaglia.

Thursday, September 21, 17:05

Theodora Bourni (UTK)

Title: Ancient pancakes I

Abstract: We show that, up to rigid motions, there is a unique compact, convex, rotationally symmetric, ancient solution of mean curvature flow that lies in a slab of width $\pi$ and in no smaller slab. This is joint work with Mat Langford and Giuseppe Tinaglia.

Thursday, September 14, 17:05

Mat Langford (UTK)

Title: Ancient solutions of the mean curvature flow

Abstract: I will present a survey of existence and rigidity results for ancient solutions of mean curvature flow. In particular, I will describe recent work (with Theodora Bourni and Giuseppe Tinaglia) on the existence and uniqueness of rotationally symmetric ancient solutions which lie in a slab. We will finish by describing some interesting open problems.

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