Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization

TL;DR

This paper constructs small-area embedded Willmore tori in Riemannian three-manifolds, using geometric expansions and Lyapunov-Schmidt reduction, proving existence results under curvature conditions.

Contribution

It introduces new geometric expansions and analysis techniques to establish the existence of area-constrained Willmore tori in specific Riemannian manifolds.

Findings

Existence of embedded area-constrained Willmore tori in compact 3-manifolds with constant scalar curvature.

Existence of such tori in the double Schwarzschild space.

Development of geometric expansions for small symmetric Clifford tori.

Abstract

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent M\"obius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behavior of degenerating tori under the action of the M\"obius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.