Eternal forced mean curvature flows III - Morse homology

TL;DR

This paper develops the Morse homology framework for forced mean curvature flows, analyzing their asymptotic behavior as forcing increases, enabling explicit calculations and revealing the existence of numerous convex hyperspheres with prescribed mean curvature.

Contribution

It completes the theoretical foundation for Morse homology in forced mean curvature flows and describes their asymptotic behavior as forcing tends to infinity.

Findings

Morse homology for forced mean curvature flows can be explicitly computed.

As forcing increases, the flows exhibit predictable asymptotic behavior.

Existence of at least 2^{d+1} convex hyperspheres with prescribed mean curvature on flat tori.

Abstract

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing term tends to infinity in a certain manner. This result allows the Morse homology to be explicitely calculated, and will permit us to show in forthcoming work that, for a large family of smooth positive functions, $F$, defined over a $(d+1)$-dimensional flat torus, there exist at least $2^{d+1}$ distinct, locally strictly convex, Alexandrov-embedded hyperspheres of mean curvature prescribed at every point by $F$.