The biharmonic hypersurface flow and the Willmore flow in higher dimensions

2025/05/26 by Yuguang Fu, Fu, Yu, Min-Chun Hong +3
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2505.19727

openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The biharmonic flow of hypersurfaces Mⁿ immersed in the Euclidean space \mathbb Rⁿ⁺¹ for n≥ 2 is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon-Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem in \citeBWW on the biharmonic hypersurface flow for n=4. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions.

Citations

• Biharmonic and biconservative hypersurfaces in space forms
• A canonical neighborhood theorem for the mean curvature flow in higher codimension
• Concentration-compactness and finite-time singularities for Chen's flow
• The mean curvature flow of submanifolds of high codimension
• Partial Regularity of a minimizer of the relaxed energy for biharmonic maps
• A Compactness Property for Solutions of the Ricci Flow
• Minimal Varieties in Riemannian Manifolds
• Harmonic Mappings of Riemannian Manifolds

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• On H²-gradient Flows for the Willmore Energy
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