Variational Theory and Spectral Theory of the Volume Functional

University Of Chicago, Chicago IL

Investigators

Abstract

Award: DMS 1710846, Principal Investigator: Andre Neves The study of shapes that are in an equilibrium position was started by Lagrange 300 year ago. These special shapes are called minimal surfaces and they are ubiquitous in Science, serving to model soap films, black holes in General Relativity, or tensile structures in architecture. Moreover, they come in two types: those which are stable, i.e., if perturbed they return to their original equilibrium position and those which are unstable, i.e., if perturbed they move away from their original position. The first type has been extensively studied in Mathematics over the last 40 years and they have been used to solve many long standing open questions in Geometry and Mathematical Relativity. The second type started being studied more systematically some years ago by Marques and the PI and they were used to solve some open problems such as the Willmore Conjecture or the problem of finding infinitely many unstable minimal surfaces. The project presented plans to continue the study of unstable minimal surfaces. Its goals are twofold: On one hand to develop the theory that governs their existence and on the other hand to study their properties as the degrees of instability become quite large because this is expected to uncover new relations across Geometry, Topology, and Analysis. More precisely, the objectives of the project are to further develop the Almgren-Pitts Min-max Theory and to study the properties of minimal surfaces when seen as nonlinear eigenvalues to the volume spectrum. For the first part we aim to investigate how to bound from below the index of min-max minimal surfaces by the number of parameters and to relate that problem with the multiplicity one conjecture. An application of that result would be to solve a stronger version of a conjecture of Yau. The second objective is motivated by the recent Weyl Law for the volume spectrum that the PI proved with Marques and Liokumovich. This property suggests a stronger analogy between eigenfunctions for the Laplacian spectrum and minimal surfaces that we intend to explore and the possibility to prove Weyl Laws for many other nonlinear problems. The techniques involved combine ideas from Spectral Geometry, Morse Theory, and Minimal Surface theory.