“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.

 

 

 

Please, visit our   SITE OF THE SEMINAR  for more information on the next seminars, organizers, etc..