Analysis Seminar

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Lower Bounds for Nodal Sets of Laplacian Eigenfunctions Using Heat Flow Arguments

Speaker: Stefan Steinerberger, Yale University

Location: Warren Weaver Hall 1302

Date: Thursday, April 23, 2015, 11 a.m.

Synopsis:

If u is a Laplacian eigenfunction on a manifold, then there are easy heuristic arguments predicting the size of {x:u(x) = 0} (in two dimensions, this set can be created using a metal plate, sand and a violin bow and has already amazed Napoleon). These heuristic arguments, though almost certainly accurate, are difficult to make rigorous. We survey existing results and give a new proof for the currently best known lower bound (proved independently by Colding & Minicozzi as well as Sogge & Zelditch). Our argument exploits the fact that the heat equation with the eigenfunction as initial value can (a) be explicitely solved for trivial reasons and (b) studied using the stochastic interpretation based on Brownian motion. We also describe some other applications of our argument.