Previous month Previous day Next day Next month
By Year By Month By Week Today Search Jump to month
“Online Analysis and PDE”

 “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..