Calculus of Variations and Geometric Measure Theory
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Quantitative and qualitative control of minimal hypersurfaces with bounded index and area

Ben Sharp

created by malchiodi on 03 Nov 2015

5 nov 2015 -- 11:00   [open in google calendar]

Scuola Normale Superiore, Aula Tonelli

Abstract.

Due to the work of Almgren-Pitts (and Schoen-Simon) we know that there exists at least one closed minimal embedded hypersurface in any closed Riemannian manifold; moreover a conjecture of Yau suggests that there should be infinitely many. This has been confirmed recently by the work of Marques-Neves in the case that the ambient manifold has positive Ricci curvature. There are therefore many open problems concerned with understanding the space of minimal hypersurfaces in a Riemannian manifold.

In this talk we will discuss some compactness theorems for minimal hypersurfaces with bounded index and area; in particular we prove that the total curvature is quantised by the limit hypersurface and finitely many properly embedded minimal hypersurfaces in Euclidean space. Easy corollaries of this result lead to qualitative estimates on topology and total curvature for minimal hypersurfaces with bounded index and area. Most of the work we present is joint with Reto Müller.

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