Spectral Selection and Minimal Morse Structures on the Poincar\'e Dodecahedral Space

TL;DR

This paper investigates the long-term behavior of the heat equation on the Poincaré dodecahedral space, showing how spectral properties influence the emergence of minimal Morse functions in solutions.

Contribution

It introduces a spectral selection property, demonstrates its failure for the round metric, and constructs nearby metrics where the property holds, linking representation theory and Morse structures.

Findings

Property P fails for the round metric.

Constructed metrics close to the round metric satisfy property P.

Spectral mechanisms connect eigenvalue splitting with Morse structures.

Abstract

We study the long time behavior of the heat equation on the spherical Poincare dodecahedral space and introduce a spectral selection property P, asserting that for a dense open set of initial data, the solution eventually becomes a minimal Morse function. We first establish an obstruction principle. If the first positive eigenspace of the Laplace Beltrami operator contains a Morse function that is not minimal, then property P fails. Using an explicit representation theoretic description of the spherical first eigenspace, we show that the round metric on M violates property P. We then develop a perturbative spectral selection mechanism. Using conformal variations and a finite dimensional reduction of the first-order splitting of the lowest eigenvalue cluster, we construct metrics arbitrarily close to the spherical metric for which the first eigenvalue is simple and the corresponding eigenfunction is minimal Morse with exactly six critical points. As a consequence, these nearby metrics satisfy property P. This establishes both the failure and the restoration of minimal Morse selection on M, and provides a concrete spectral mechanism linking representation theory, eigenvalue splitting, and global Morse structure.