## University of Tennessee - Geometric analysis seminar

### Fall 2019

Thursday, October 3, 16:00, A113

Li Chen (UConn)

Title: BV functions and isoperimetry on Dirichlet spaces.

Abstract: I present a theory of BV functions on Dirichlet spaces, focusing on the strictly local case with Gaussian heat kernel bounds and the strongly local case with sub-Gaussian heat kernel bounds. Under weak Bakry–Emery curvature type conditions, we prove global isoperimetric inequalities.

Thursday, October 10, 16:00, A113

Florian Johne (Columbia)

Title: Surgery for an extended Ricci flow system

Abstract: *List flow* is a geometric flow for a pair (*g,u*), where *g* is a Riemannian metric and *u* a smooth function. This extended Ricci flow system has applications to static vacuum solutions of the Einstein equations and to Ricci flow on warped products.
The coupling in this flow induces additional difficulties compared to Ricci flow, which we overcome by proving an improved bound on the Hessian. This allows us to prove a convergence result, a singularity classification result and a surgery result in three dimensions.

Thursday, October 24, 16:00, A113

Alex Mramor (Johns Hopkins)

Title: Ancient solutions to mean curvature flow

Abstract: In this talk I will discuss joint work with Alec Payne on a construction of ancient solutions to mean curvature flow coming out of minimal surfaces.

Thursday, November 21, 16:00, A113

Julian Scheuer (Columbia)

Title: Isoperimetric problems in Lorentzian manifolds

Abstract: The classical isoperimetric and Minkowski inequalities in the Euclidean space relate the enclosed volume, the surface area and the total mean curvature of certain hypersurfaces. In this talk we present a curvature flow approach to prove properly defined analogues in certain classes of Lorentzian manifolds.

Thursday, November 21, 16:00, A113

Xuan Hien Nguyen (Iowa State)

Title: The fundamental gap of convex domains in *H*^{2}

Abstract: We give a brief history of progress on fundamental gap problem in *R*^{n} and *S*^{n}. Then we compute the fundamental gap of a family of convex domains in the hyperbolic plane *H*^{2} showing that there are convex domains for which *λ*_{1} – λ_{2} ≤ 3 π^{2}/D^{2}, where *D* is the diameter of the domain and *λ*_{1}, *λ*_{2} are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with the case of domains in *R*^{n} or *S*^{n}, where *λ*_{2} – λ_{1} ≥ 3 π^{2}/D^{2}.

Thursday, December 5, 16:00, A113

Raquel Perales (Universidad Nacional Autonoma de Mexico)

Title: Stability of graphical tori with almost nonnegative scalar curvature.

Abstract: By works of Schoen–Yau and Gromov–Lawson, any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to zero. We prove flat and intrinsic flat subconvergence to a flat torus for sequences of 3-dimensional tori *M*_{j} that can be realized as graphs of functions defined over flat tori satisfying a uniform upper diameter bound, a uniform lower bound on the area of the smallest closed minimal surface, and scalar curvature bounds of the form *R*_{gMj} ≥ -1/j. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang–Lee and Huang–Lee–Sormani.

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