High Frequency Eigenfunction Immersions and Supremum Norms of Random Waves

TL;DR

This paper explores how high frequency eigenfunctions can immerse Riemannian manifolds into Euclidean space, analyzing the geometry and providing new bounds on the maximum values of random eigenfunction combinations.

Contribution

It introduces a novel geometric perspective on eigenfunction immersions and offers a new proof for existing bounds on eigenfunction supremum norms.

Findings

Manifolds can be immersed into Euclidean space via high frequency eigenfunctions.

Provides a new proof for upper bounds of supremum norms of random eigenfunction combinations.

Enhances understanding of the geometry induced by eigenfunctions on manifolds.

Abstract

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.