The Stable Bernstein theorem in R^5

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• Paul Minter (Clay Institute, Princeton, Cambridge)
• Friday 19 January 2024, 14:00-15:00
• MR15.

If you have a question about this talk, please contact Dr Greg Taujanskas.

Short Version: I will explain why immersed stable minimal hypersurfaces in R5 are flat. This is joint work with Otis Chodosh, Chao Li, and Douglas Stryker.

Long Version: The stable Bernstein problem asks whether an immersed stable minimal hypersurface in Rn is necessarily flat. If true the result implies, for example, a priori curvature estimates for immersed stable minimal hypersurfaces in Riemannian n-manifolds.

An important special case of the result, known as the Bernstein problem, asks the same question except for minimal graphs over a hyperplane (such graphs are in fact locally area minimising). The Bernstein problem was resolved in full in the 1960’s by works of Fleming, De Giorgi, Almgren, Simons, Bombieri, and Giusti, in turn driving a lot of progress in the development of geometric measure theory. It was shown that such a minimal graph must be flat if it lies in R8 or lower, whilst counterexamples exist in R9 and higher. The stable Bernstein theorem has counterexamples in R8 and above for related reasons.

The stable Bernstein theorem in its full generality remained essentially open until recently. The works of Schoen—Simon—Yau (1975) and Bellettini (2023) establish the result up to R7 assuming the minimal hypersurface has Euclidean volume growth. The full R3 case was resolved independently by works of Fischer-Colbrie—Schoen, do Carmo—Peng, and Pogorelov, all around 1980. In 2021, Chodosh—Li proved the stable Bernstein theorem in R4, using techniques from non-negative scalar curvature. They later found another proof using techniques from uniformly positive scalar curvature utilising Gromov’s mu-bubbles, showing that the Euclidean volume growth assumption holds a priori.

In this talk, I will discuss recent work (joint with Otis Chodosh, Chao Li, and Douglas Stryker) resolving the stable Bernstein problem in R5. Our proof uses instead techniques from the study of uniformly positive bi-Ricci curvature. We show that a suitable conformal change of metric (the Gulliver—Lawson metric) has uniformly positive bi-Ricci curvature in a spectral sense. This allows us to construct (warped) mu-bubbles with uniformly positive Ricci curvature in a spectral sense, from which we can establish a Bishop volume-comparison theorem (which is a subtle adaptation of a technique from Bray; indeed, our proof seems to break down in any higher dimensional situation). These ingredients allow us to establish a priori Euclidean volume growth for the minimal hypersurface allowing us to prove the result.

This talk is part of the Partial Differential Equations seminar series.

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