Introducing spontaneous curvature to the Helfrich flow: Singularities and convergence

TL;DR

This paper investigates the long-term behavior of the Helfrich flow with spontaneous curvature, revealing conditions for finite-time singularities and convergence, and providing explicit energy thresholds based on curvature and constraints.

Contribution

It introduces the analysis of Helfrich flow with spontaneous curvature, establishing conditions for singularities and convergence, and deriving explicit energy thresholds for global existence.

Findings

Negative spontaneous curvature leads to finite-time singularities near spheres.

Positive spontaneous curvature ensures global existence and convergence under small energy.

Explicit energy thresholds depend on spontaneous curvature and area constraints.

Abstract

While there are various results on the long-time behavior of the Willmore flow, the Helfrich flow with non-zero spontaneous curvature as its natural generalization is not yet well-understood. Past results for the gradient flow of a locally area- and volume-constrained Willmore flow indicate the existence of finite-time singularities which corresponds to the scaling-behavior of the underlying energy. However, for a non-vanishing spontaneous curvature, the scaling behavior is not quite as conclusive. Indeed, in this article, we find that a negative spontaneous curvature corresponds to finite-time singularities of the locally constrained Helfrich flow if the initial surface is energetically close to a round sphere. Conversely however, in the case of a positive spontaneous curvature, we find a positive result in terms of the convergence behavior: The locally area-constrained Helfrich flow starting in a spherical immersion with suitably small Helfrich energy exists globally and converges to a Helfrich immersion after reparametrization. Moreover, this energetic smallness assumption is given by an explicit energy threshold depending on the spontaneous curvature and the local area constraint of the energy.