2013/06/25 by Yong Luo, Luo, Yong · 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
paper · pdf · doi:10.48550/arxiv.1306.6069
openalex publication_date 2013/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the biharmonic submanifolds theory there is a generalized Chen's conjecture which states that biharmonic submanifolds in a Riemannian manifold with non-positive sectional curvature must be minimal. This conjecture turned out false by a counter example of Y. L. Ou and L. Tang in \citeOu-Ta. However it remains interesting to find out sufficient conditions which guarantee this conjecture to be true. In this note we prove that: 1. Any complete biharmonic submanifold (resp. hypersurface) (M, g) in a Riemannian manifold (N, h) with non-positive sectional curvature (resp. Ricci curvature) which satisfies an integral condition: for some p∈ (0, +∞), ∫M|H|pdug0 which satisfies that ∫Bρ(x0)|H|p+2dμg(p≥0) is of at most polynomial growth of ρ, must be minimal. We also consider ε-superbiharmonic submanifolds defined recently in \citeWh by G. Wheeler and prove similar results for ε-superbiharmonic submanifolds, which generalize the result in \citeWh.