First-eigenvalue maximization and inflation of maps

TL;DR

This paper explores optimization problems related to the first eigenvalue of the Bakry-Emery Laplacian and smooth maps into Hilbert space on compact manifolds, providing explicit solutions and a Nadirashvili-type theorem.

Contribution

It introduces a dual formulation of eigenvalue optimization and map inflation problems on manifolds, with explicit examples and a new theoretical result.

Findings

Explicit solutions for certain manifolds

A Nadirashvili-type theorem proved

Dual optimization problems formulated and analyzed

Abstract

Given a compact manifold equipped with a volume element and a Riemannian metric, we formulate and study a dual pair of optimization problems: one concerning smooth maps from the manifold into the Hilbert space $l^2$ and the other concerning the smallest positive eigenvalue of the Bakry-Emery Laplacian. We present examples of manifolds for which these problems can be solved explicitly. We also prove a Nadirashvili-type theorem.