University of Tennessee - Geometric analysis seminar

The seminar will be run via Zoom (ID: 912 2826 6699).

Please contact tbourni at utk dot edu to sign up for the mailing list.

Spring 2022

Tuesday, March 3, 1450-1550
Ao Sun (University of Chicago)
Title: Existence of minimal hypersurfaces with arbitrarily large area.
Abstract: I will present an approach to find minimal hypersurfaces with arbitrarily large area in a closed manifold with dimension between 3 and 7. The method is based on the novel Almgren–Pitts min-max theory, and its further development by Marques–Neves, Song and Zhou. Among the applications, we can show that there exist minimal hypersurfaces with arbitrarily large area in an analytic manifold. In the case where this approach does not work, it is surprising that the space of minimal hypersurfaces has a Cantor set fractal structure. This is joint work with James Stevens (UChicago).

Tuesday, March 8, 1450-1550
Mariel Sáez (Pontificia Universidad Católica de Chile)
Title: Uniqueness of entire graphs evolving by mean curvature flow.
Abstract: In this talk I will discuss the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general n-dimensional case, we prove uniqueness under additional conditions: we require a uniform lower bound on the second fundamental form and the height function. The latter result extends to initial conditions that are proper graphs over subdomains of Rn. (Joint with P. Daskalopoulos).

Tuesday, March 29, 1450-1550 (Ayr 122)
Brett Kotschwar (Arizona State University)
Title: Backward propagation of warped-product structures under the Ricci flow and asymptotically conical shrinkers.
Abstract: We establish sufficient conditions for a locally-warped product structure to propagate backward in time under the Ricci flow. As an application, we show that if a gradient shrinking soliton is asymptotic to a cone whose cross-section is a locally warped product of Einstein manifolds, the soliton must itself be a warped product over the same manifolds.

Tuesday, April 5, 1450-1550 (Ayr 122)
Pedro Gaspar Marques Da Silva (University of Chicago)
Title: Phase transitions and mean curvature flows in the sphere.
Abstract: The Allen–Cahn equation is a semilinear evolution PDE that models phase transition phenomena. Since the eighties, this equation has been used as a regularization for the mean curvature flow (MCF), providing a useful tool to construct and study solutions of this geometric flow. In this talk, we will discuss an existence result for eternal solutions to the Allen–Cahn equation on a round sphere connecting unstable equilibria, and show that it can be used to construct a family of (weak) eternal solutions to the MCF connecting minimal surfaces of low area in the 3-sphere. This is joint work with Jingwen Chen.

Tuesday, April 19, 1200-1300
Keaton Naff (MIT)
Title: Immersed mean convex mean curvature flows with noncollapsed singularities.
Abstract: In the mean curvature flow of hypersurfaces, noncollapsing has proven to be a powerful and useful assumption when studying singularities and high curvature regions. In particular, the assumption of noncollapsing has been used to prove a wide range of local a priori estimates, and has led to classification results for certain classes of singularity models. Less is known for immersed mean-convex flows. In this talk, I would like to survey recent results and discuss outstanding conjectures for immersed mean-convex flows that begin to bridge the gap between the embedded and immersed mean-convex settings. The talk is based on joint work with S. Brendle and ongoing work with S. Lynch.

Tuesday, April 26, 1450-1550
Elena Mäder-Baumdicker (Technische Universität Darmstadt)
Title: The area preserving curve shortening flow in a free boundary setting.
Abstract: A convex, simple closed plane curve moving by the area preserving curve shortening flow (APCSF) converges smoothly to a circle with the same enclosed area as the initial curve (Gage 1986). Note that the circle is the solution of the isoperimetric problem in the Euclidean plane. Corresponding to the relative (outer) isoperimetric problem we present results concerning the APCSF with Neumann free boundary conditions outside of a convex domain. Under certain conditions on the initial curve the flow does not develop a singularity and subconverges smoothly to an arc of a circle sitting outside of the given convex domain and enclosing the same area as the initial curve. On the other hand, there are many examples of convex initial curves developing a singularity in finite time. In all these cases, the singularity is of type II, and we conjecture that some curves developing a singularity stay embedded under the flow. In general, we will point out similarities and differences of the APCSF to the well-known curve shortening flow.

Tuesday, April 26, 1450-1550
Yi Lai (Stanford University)
Title: O(2)-symmetry of 3D steady gradient Ricci solitons
Abstract: For any 3D steady gradient Ricci soliton, if it is asymptotic to a ray we prove that it must be isometric to the Bryant soliton. Otherwise, it is asymptotic to a sector and called a flying wing. We show that all flying wings are O(2)-symmetric. Hence, all 3D steady gradient Ricci solitons are O(2)-symmetric.

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