Adiabatic theory for the area-constrained Willmore flow

TL;DR

This paper develops an adiabatic theory to approximate the evolution of large surfaces under the area-constrained Willmore flow in a Schwarzschild manifold, using an explicit barycenter map to reduce the problem to effective dynamics.

Contribution

It introduces a novel explicit map for barycenters that captures static and dynamic properties of the flow, enabling a simplified effective dynamics approximation.

Findings

Derived an explicit four-dimensional effective barycenter dynamics.

Constructed surfaces closely following prescribed barycenter flows.

Provided conditions under which the approximation remains valid over time.

Abstract

In this paper, we develop an adiabatic theory for the evolution of large closed surfaces under the area-constrained Willmore (ACW) flow in a three-dimensional asymptotically Schwarzschild manifold. We construct explicitly a map, defined on a certain four-dimensional manifold of barycenters, which characterizes key static and dynamical properties of the ACW flow. In particular, using this map, we find an explicit four-dimensional effective dynamics of barycenters, which serves as a uniform asymptotic approximation for the (infinite-dimensional) ACW flow, so long as the initial surface satisfies certain mild geometric constraints (which determine the validity interval). Conversely, given any prescribed flow of barycenters evolving according to this effective dynamics, we construct a family of surfaces evolving by the ACW flow, whose barycenters are uniformly close to the prescribed ones on a large time interval (whose size depends on the geometric constraints of initial configurations).