The principal eigenfunction of the Dirichlet Laplacian with prescribed numbers of critical points on the upper half of a topological torus

TL;DR

This paper constructs a specific topological torus's upper half to produce a principal eigenfunction with a prescribed number of critical points and explicitly identifies their locations.

Contribution

It introduces a novel construction method for topological tori that yields eigenfunctions with exact critical point counts and explicit critical point locations.

Findings

Constructed a topological torus with desired critical points

Explicitly identified all critical point locations

Demonstrated control over eigenfunction critical point structure

Abstract

We consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.