Geometric Analysis and Topology Seminar

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Existence and regularity of anisotropic minimal hypersurfaces

Speaker: Yangyang Li, University of Chicago

Location: Warren Weaver Hall 512

Date: Friday, March 15, 2024, 11 a.m.

Synopsis:

Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. The regularity theory for its critical points, anisotropic minimal hypersurfaces, is significantly more challenging than the area functional case, mainly due to the absence of a monotonicity formula. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface and optimally regular minimal hypersurfaces for elliptic integrands in closed Riemannian manifolds through min-max construction. This confirms a conjecture by Allard in 1983. If time permitted, I will also discuss how this could be connected to the minimal surface theory. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.