Geometric Analysis and Topology Seminar

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Minimal hypersurfaces with variable asymptotic directions and the Bernstein problem

Speaker: Jinxin Xue, Tsinghua University

Location: Warren Weaver Hall 512

Date: Friday, October 30, 2026, 11 a.m.

Synopsis:

Geodesics dynamics on nonflat 2-torus is very complicated. When considering globally length minimizing geodesics, there is a good theory developed since Birkhoff, Morse, Hedlund and later re-incarnated by Aubry and Mather. The 2D result admits a higher dimensional generalization by Moser and Bangert, who study non self-intersecdting minimal hypersurfaces. There was a remarkable result of Birkhoff which implies the existence of complete geodesics with different asymptotic directions. We show that this also admits a higher dimensional generalization to minimal surfaces, which implies in particular, minimal hypersurfaces with different asymptotic directions exist for generic small periodic perturbation of the flat metric. This shows that the Bernstein problem is unstable under such perturbations.