A remark on the Heat Equation and minimal Morse Functions on Tori and Spheres

TL;DR

This paper investigates whether solutions to the Heat Equation on flat tori and round spheres tend to minimal Morse functions over time, providing initial evidence for this behavior on these specific manifolds.

Contribution

It demonstrates that for flat tori and round spheres, solutions to the Heat Equation become minimal Morse functions for large times, advancing understanding of generic initial conditions on these manifolds.

Findings

Solutions tend to minimal Morse functions on tori and spheres

Behavior observed in all dimensions for these manifolds

Supports conjecture for homogeneous manifolds

Abstract

Let (M,g) be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of points p,q in M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each "generic" initial condition f0, the solution to the Heat Equation is such that for sufficiently large t, f(.,t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.