Thursday, April 26, 17:05

Sajjad Lakzian (Fordham)

Title: TBA.

Abstract: TBA.

Thursday, April 19, 17:05

Ling Xiao (UConn)

Title: TBA.

Abstract: TBA.

Thursday, March 29, 17:05

Martin Reiris (CMAT)

Title: TBA.

Abstract: TBA.

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.