Analysis and PDEs Seminar

Nonlocal approximation of minimal surfaces: optimal estimates from stability

Speaker:  Joaquim Serra (ETH Zurich)
Date:  Wednesday, 17 May 2023 - 15:00


Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren- Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture.

In a recent paper with Chan, Dipierro, and Valdinoci we set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that stable s-minimal surfaces in the unit ball of \(R^3\) satisfy curvature estimates that are robust as s approaches 1 (i.e. as the energy approaches that of classical minimal surfaces). Moreover, we obtain optimal sheet separation estimates and show that critical interactions are encoded by nontrivial solutions to a  (local) "Toda type" system. As a nontrivial application, we establish that hyperplanes are the only stable s-minimal hypersurfaces in \(R^4\), for \(s\) sufficiently close to 1.

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