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“Online Analysis and PDE”
Wednesday May 17th at 15:00h.
Speaker: Joaquim Serra
ETH Zurich
Title: Nonlocal approximation of minimal surfaces: optimal estimates from stability.
Abstract: 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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