University of Tennessee - Geometric analysis seminar
Spring 2018
Thursday, April 26, 17:05
Sajjad Lakzian (Fordham)
Title: Compactness theory for harmonic maps from Riemann surfaces into compact locally CAT(1) spaces.
Abstract: In this talk, we will show the bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost and Parker respectively established these results by exploiting now classical arguments for harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. In the absence of a PDE, we take advantage of the local convexity properties of the target space. Included in this paper are an ε-regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image. If time allows, we will also touch upon other ideas and developments regarding the existence of such maps in free homotopy classes. This is a joint work with C. Breiner.
Thursday, April 19, 15:00 in A404 (Note time and room change)
Ling Xiao (UConn)
Title: Complete translating solitons to the mean curvature flow in $R^3$ with nonnegative mean curvature.
Abstract: We prove that any complete immersed two-sided mean convex translating soliton $\Sigma\subset R^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $R^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $\Sigma$ tends to zero at infinity then $\Sigma$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions. This is a joint work with Joel Spruck
Tuesday, April 10, 17:05 (General relativity seminar)
Henri Roesch (UC Irvine)
Title: Isolated Horizons and the Null Penrose Inequality.
Abstract: In the first half of the talk, we introduce a new quasi-local mass with interesting properties along null flows off of a 2-sphere in spacetime or, equivalently, foliations of a null cone. We also show how certain, fairly generic, convexity assumptions on the null cone allows for a proof of the Null Penrose Inequality. On the Black Hole Horizon, we find that the convexity assumptions become sharp; therefore, the second half of the talk will explore the existence of a class of Black Hole Horizons admitting such convexity. From this, building upon the work of S. Alexakis, we will show that the Schwarzschild Null Cone--the case of equality for the Penrose Inequality--is also stable under small metric perturbations.
Thursday, March 29, 17:05
Martin Reiris (CMAT)
Title: Bakry-Émery geometric comparison techniques and applications to general relativity.
Abstract: Starting with the work of Bakry, Émery, Ledoux, and others on diffusion processes on Riemannian manifolds, several novel Riemannian comparison techniques with broad applications have been developed over the last decades. Classical theorems, like Myer's compactness or Cheeger-Gromoll's splitting were generalised and impressive applications were found by Perelman in the Ricci flow and soliton theory. In this talk I will begin reviewing the probabilistic origins of such techniques, introduce then some basic theorems, and finally show applications to a wide range of problems related to static and stationary solutions of the Einstein equations (the 'solitons' of the theory) with or without matter. Open problems and prospective directions of work will be mentioned.
Thursday, March 22, 17:05
Kyeongsu Choi (MIT)
Title: Free boundary problems in the Gauss curvature flow.
Abstract: We will discuss about the optimal $C^{1,1/(n-1)}$ regularity of the Gauss curvature flow with a flat side. We will consider several quantities which are degenerate or singular near the flat side, and establish estimates for their ratios. Geometric meaning of the ratios will be discussed. Moreover, by using the ratios, we will classify the closed self-similar solutions to the Gauss curvature flow.
Thursday, March 8, 17:05
Brian Allen (USMA)
Title: Contrasting notions of convergence in geometric analysis.
Abstract: Often times when one studies sequences of Riemannian manifolds, which arise naturally when studying stability questions, one arrives at convergence in an $L^p$ space on the way to achieving Gromov-Hausdorff (GH) or Intrinsic Flat (IF) convergence. This motivates the desire to find conditions which when combined with $L^p$ convergence imply GH and/or IF convergence. In this talk we will discuss a Theorem which identifies such conditions and look at some recent applications to stability questions from geometric analysis which take advantage of these insights. This is joint work with Christina Sormani.
Thursday, March 1, 17:05
Ryan Unger (UTK)
Title: The isoperimetric problem and spherical rearrangements II.
Abstract: We show how to decrease the Sobolev norm of a smooth function by spherically rearranging its level sets (Pólya-Szegő inequality). This is connected to the Euclidean isoperimetric problem and the best constant in the Sobolev inequality.
Thursday, February 22, 17:05
Ryan Unger (UTK)
Title: The isoperimetric problem and spherical rearrangements.
Abstract: We show how to decrease the Sobolev norm of a smooth function by spherically rearranging its level sets (Pólya-Szegő inequality). This is connected to the Euclidean isoperimetric problem and the best constant in the Sobolev inequality.
Thursday, February 9, 17:05
William H. Meeks III (University of Massachusetts Amherst)
Title: Progress in the theory of CMC surfaces in locally homgeneous 3-manifolds.
Abstract: I will go over some recent work that I have been involved in on surface geometry in complete locally homogeneous 3-manifolds, X. In joint work with Mira, Perez and Ros, we have been able to finish a long term project related to the Hopf uniqueness/existence problem for CMC spheres in any such X. In joint work with Tinaglia on curvature and area estimates for CMC H>0 surfaces in such an X, we have been working on getting the best curvature and area estimates for constant mean curvature surfaces in terms of their injectivity radii and their genus. It follows from this work that if W is a closed Riemannian homology 3-sphere then the moduli space of closed embedded surfaces of constant mean curvature H in an interval [a,b] with a>0 and of genus bounded above by a positive constant is compact. In another direction, in joint work with Coskunuzer and Tinaglia, we now know that, in complete hyperbolic 3-manifolds N, any complete embedded surface M of finite topology is proper in N if H is at least 1 (this is work with Tinaglia) and for any value of H less than 1 there exists complete embedded nonproper planes in hyperbolic 3-space (joint work with both researchers). In joint work with Adams and Ramos, we have been able characterize the topological types of finite topology surfaces that properly embed in some complete hyperbolic 3-manifold of finite volume (including the closed case) with constant mean curvature H; in fact, the surfaces that we construct are totally umbilic.
Thursday, January 25, 17:05
Lan-Hsuan Huang (University of Connecticut)
Title: Recent progress on the positive mass theorem.
Abstract: The positive mass theorem in general relativity asserts that the Arnowitt--Deser--Misner (ADM) mass of an asymptotically flat manifold satisfying the dominant energy condition is nonnegative. Furthermore, if the ADM mass is zero, then the manifold is a slice of Minkowski spacetime. A very important special case, the so-called the Riemannian positive mass theorem, was first proved by R. Schoen and S.T. Yau in the 1980's. Later, E. Witten gave a different proof of the general case, under a topological condition that the manifold is spin. We will discuss our recent results that hold in greater generality and remove the spin condition.
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