Heat equation and stable minimal Morse functions on real and complex projective spaces

TL;DR

This paper proves that solutions to the heat equation on real and complex projective spaces evolve to stable minimal Morse functions regardless of initial conditions, extending previous results on flat tori and spheres.

Contribution

It establishes that on real and complex projective spaces, heat equation solutions become stable minimal Morse functions over time, generalizing earlier findings to new geometric settings.

Findings

Solutions become minimal Morse functions over time

Solutions stabilize regardless of initial conditions

Extends previous results to projective spaces

Abstract

Following similar results in arXiv:1301.5934 for flat tori and round spheres, in this paper is presented a proof of the fact that, for "arbitrary" initial conditions $f_0$, the solution $f_t$ at time $t$ of the heat equation on real or complex projective spaces eventually becomes (and remains) a minimal Morse function. Furthermore, it is shown that the solution becomes stable.