Cornell Analysis Seminar

Title:  Existence of constant mean curvature surfaces with bounded genus in general 3-manifolds. 

Abstract: In this talk, I will introduce some history on finding minimal surfaces and constant mean curvature (CMC) surfaces. Then, I will talk about our recent work on the construction of CMC surfaces with genus control in a general 3-manifold. Our construction is based on a penalized area functional introduced by Riviere. This talk is based on joint work with Filippo Gaia. 

Title: Minimal hypersurfaces of Morse index one

Abstract: I will discuss joint work with Otis Chodosh in which we prove that the catenoid is the unique embedded minimal hypersurface of Morse index one in R^n.

Title: Min-max barrier and minimal foliations on the torus. 

Abstract: Aubry–Mather theory concerns minimizing orbits of convex Hamiltonian systems. Moser, Bangert, and Auer developed a higher-dimensional analogue for area-minimizing hypersurfaces. In this talk, I will describe this theory on a Riemannian torus. After lifting to the universal cover, each minimizing hypersurface carries an asymptotic invariant, called its homological direction, and hypersurfaces with a common direction organize into laminations. I will discuss recent work in which Almgren–Pitts min–max theory is used to characterize when such a lamination is a foliation. I will also explain how, for generic metrics, a gap in the lamination contains a complete embedded minimal hypersurface that is not area-minimizing. 

Title: The Positive Mass Theorem in Arbitrary Dimensions 

Abstract: We present an inductive scheme for proving the positive mass theorem in arbitrary dimensions, extending the Schoen–Yau argument from dimensions at most 7 to the general case. This is a joint work with Simon Brendle. 

Title: Positive Scalar curvature and volume growth.

Abstract: In 1985, Gromov proposed several conjectures about large scale topological and geometric consequences of uniformly positive scalar curvature. They all reflect that uniformly positive scalar curvature forces codimension two in large scales. One of the conjectures is that under additional nonnegative Ricci curvature condition, the volume growth of a geodesic ball of radius $r$ is at most of the order $r^{n-2}$. 

In this talk, we present (the outline of) the proof of this conjecture, highlighting the new ideas that involve heat flow, splitting and geodesic flow. This is joint work with Bochao Kong with substantial AI assistance.

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