Eventual Morse Minimality of Heat Flow on Lens Spaces

Abstract

We prove that heat flow generically produces Morse-minimal functions on round lens spaces. The result gives a rigorous explanation of a phenomenon first suggested by computational experiments: after high-frequency modes have been suppressed by diffusion, generic heat evolutions on lens spaces tend to settle into Morse functions with the smallest possible number of critical points. Precisely, for every lens space \(L(p,q)\) with \(p\geq2\), \(1\leq q\leq p/2\), and \((p,q)=1\), there is an open dense set of initial data \(f\in L^2(L(p,q),\mathbb R)\) such that the heat evolution \(e^{t\Delta}f\), where \(\Delta\) is taken in the non-positive sign convention, is, for all sufficiently large \(t\), a Morse function with exactly four critical points, of indices \(0,1,2,3\). The proof analyzes the asymptotic spectral expansion of the heat flow. In the most delicate case \(1<q<p/2\), the leading term is Morse--Bott with two critical circles, and the higher heat modes break these circles through a resonant Fourier mechanism. An arithmetic estimate shows that the fundamental reduced frequency dominates all higher multiples, and generic nonvanishing of the corresponding resonant coefficients gives the minimal critical-point count.