Biconservative hypersurfaces with constant scalar curvature in space forms

2021/10/07 by Yu Fu, Min-Chun Hong, Fu, Yu +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.2110.03438

20pages

arxiv created 2021/10/07 · openalex publication_date 2021/10/07 · arxiv updated 2021/10/08 · openalex created_date 2021/10/11 · openalex updated_date 2026/07/28

Abstract

Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces Mⁿ with constant scalar curvature in a space form Nⁿ⁺¹(c). We prove that every biconservative hypersurface with constant scalar curvature in N⁴(c) has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in N⁵(c) is ether an open part of a certain rotational hypersurface or a constant mean curvature hypersurface. These solve an open problem proposed recently by D. Fetcu and C. Oniciuc for n≤4.

Citations

• The scalar curvature of minimal hypersurfaces in spheres
• Local Rigidity Theorems for Minimal Hypersurfaces
• Minimal Varieties in Riemannian Manifolds

Cited by

• Biconservative Weingarten surfaces with flat normal bundle in N⁴ (ε)

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