Spontaneous curvature effects of the Helfrich flow: singularities and convergence

• Published in: Communications in partial differential equations Vol. 50; no. 3; pp. 441 - 476
• Main Author: Schlierf, Manuel
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.03.2025; Taylor & Francis Ltd
• Subjects: 35K41 (secondary); 49Q10 (primary); Canham-Helfrich model; Constraints; Convergence; Curvature; geometric flows; Gradient flow; Helfrich flow; Singularities; Submerging; Willmore energy; Willmore surfaces; Łojasiewicz-Simon inequality
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2025.2457047

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 correspond 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 close to a round sphere in terms of its Willmore energy. 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 from 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.