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Seminar

The Differential Geometry of Level Sets of Eigenfunctions

Speaker
David Jerison (MIT)
Date
Tue, Jan 9 2024, 4:00pm
Location
384H

Abstract

We introduce a new way to look at level sets of eigenfunctions by viewing the value of the eigenfunction as an independent time variable, with successive level surfaces evolving over time. The evolution obeys a variational principle analogous to mean curvature flow, and this suggests studying the level surfaces in analogy with the approach of Hamilton and Perelman to mean curvature flow and Ricci flow. 

We will describe the solitons of our theory and define new geometric invariants of the level surfaces, parallel to the classical extrinsic invariants of surfaces in affine differential geometry and Riemannian geometry. We expect these invariants to lead to a more precise understanding of the shapes of level surfaces.