Johns Hopkins Geometric Analysis Seminar

Title: The Smale conjecture for RP^3 and minimal surfaces

Abstract:  In the early 80s Hatcher proved the Smale Conjecture, asserting that the diffeomorphism group of the three-sphere retracts onto its isometry group.  The corresponding problem for RP^3 was open nearly 40 years, and resolved only in 2019 by a detailed study of Ricci flow with surgeries.  I will explain a new proof using minimal surfaces.  This is joint work with Yevgeny Liokumovich.

Title: On Backwards uniqueness for singular mean curvature flows.

Abstract: Mean curvature flow, the gradient flow of the area functional, is the most natural geometric heat flow for embedded hypersurfaces. Being non linear, the flow develops singularities, at which it stops being smooth. One fundamental, often delicate, question for such non linear flows is that of backwards uniqueness. In this talk I will discuss recent backwards uniqueness results, obtained jointly with Josh Daniels-Holgate, which can address some singularities. I will also compare these results to (commonly more robust) forward uniqueness results, and also to the situation in other equations.   

Title: Widths, Index, Intersection, and Isospectrality

Abstract: In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this.

Title: From almost smooth spaces to RCD spaces

Abstract: RCD spaces, including Ricci limit spaces as typical examples, provide us a best framework treating Ricci curvature lower bound on non-smooth spaces. There are so many applications of the study to other fields, including recent works by G. Szekelyhidi and his collaborators. We provide various characterizations for a given almost smooth space to be an RCD space. Applications include a characterization of Einstein 4-orbifolds. This talk is based on a joint work with Song Sun (Zhejiang University).

Title: Electrostatics for Schroedinger operators and mitosis of minimal surfaces

Abstract: I'll discuss joint work with Adrian Chu, in which we relate Kapouleas's doubling construction for minimal surfaces to the variational theory for a Coulomb-type interaction energy for Schroedinger operators. Namely, for the Jacobi operator of a given nondegenerate minimal surface, we show that families of nondegenerate critical points of this energy give rise to high-genus minimal surfaces approximating the initial surface with multiplicity two, provided a few key estimates are satisfied. By studying the ground states for this interaction energy, we show that, generically, every minimal surface of index one admits such a doubling, and deduce as a corollary that generic 3-manifolds contains sequences of embedded minimal surfaces with bounded area and arbitrarily large genus.

Title: Rigidity in General Relativity

Abstract: We discuss several rigidity phenomena arising in geometry and relativity.

Title: A conformally invariant energy for the Paneitz operator on cornered Riemannian manifolds

Abstract: We construct conformally invariant boundary and corner operators associated to the Paneitz operator on a cornered Riemannian manifold. These operators give rise to a conformally invariant energy for the Paneitz operator on a cornered Riemannian manifold, and hence to formulations of the fourth-order Yamabe Problem on cornered conformal manifolds. We give sufficient conditions to solve a representative family of these Yamabe-type problems in the Sobolev space W^{2,2}.  This is joint work with Yueh-Ju Lin, Stephen McKeown, and Cheikh Ndiaye.

Title: Stable Bernstein Problem in certain positively curvature manifolds.

Abstract: We formulate a stable Bernstein problem in two types of positively curved manifolds. The problem is completely solved for one type in all dimensions and partially solved for the other in low dimensions, leading to several interesting corollaries.

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