Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II: Morse Theory

TL;DR

This paper develops a Morse Theory approach to construct embedded Willmore tori with small area in Riemannian three-manifolds, providing new geometric expansions and conditions for existence and multiplicity based on curvature and Morse inequalities.

Contribution

It introduces a Morse Theory framework for constructing small-area Willmore tori, extending previous minimization methods and deriving new asymptotic expansions for the Willmore functional.

Findings

Established geometric expansions of the Willmore functional derivative on degenerating tori.

Derived sufficient conditions for existence and multiplicity of stationary tori based on curvature and Morse inequalities.

Provided a Morse Theory-based construction method for embedded Willmore tori in Riemannian three-manifolds.

Abstract

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the M\"obius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore functional on exponentiated small Clifford tori degenerating, under the action of the M\"obius group, to small geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.