Location of hot spots in thin curved strips

TL;DR

This paper investigates the location of extremal points of Neumann eigenfunctions in thin curved strips, proving the hot spots conjecture for a broad class of possibly non-convex, non-Euclidean domains.

Contribution

It establishes a link between eigenfunction extrema in thin tubular neighborhoods and their underlying curves, proving the hot spots conjecture in new domain classes.

Findings

Maxima and minima of eigenfunctions are located in relation to the underlying curves.

The hot spots conjecture is proved for a large class of non-convex, non-Euclidean domains.

Eigenfunction extremal points are characterized in thin curved strips.

Abstract

The maxima and minima of Neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of Neumann eigenfunctions of the underlying curves. In particular, the hot spots conjecture for a new large class of domains (possibly non-convex and non-Euclidean) is proved.