Generalized Minimizing Movements for the varifold Canham-Helfrich flow

TL;DR

This paper develops a new mathematical framework for analyzing the evolution of surfaces under the Canham-Helfrich energy, proving existence and bounds for solutions in a varifold setting and establishing key geometric estimates.

Contribution

It introduces a generalized minimizing movements approach for the Canham-Helfrich flow and proves existence of solutions in Wasserstein spaces of varifolds, including regularity and conservation properties.

Findings

Existence of solutions in Wasserstein spaces of varifolds.

Upper and lower diameter bounds for evolving surfaces.

Li-Yau-type estimate and conservation of multiplicity in regular settings.

Abstract

The gradient flow of the Canham-Helfrich functional is tackled via the Generalized Minimizing Movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered $C^{1,1}$ surfaces, we provide a Li-Yau-type estimate for the Canham-Helfrich energy and prove the conservation of multiplicity along the evolution.