L²-harmonic p-forms on submanifolds with finite total curvature

2018/03/29 by Jundong Zhou, Zhou, Jundong
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1803.11468

openalex publication_date 2018/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H^p(L²(M)) be the space of all L²-harmonic p-forms (2≤ p≤ n-2) on complete submanifolds M with flat normal bundle in spheres. In this paper, we first show that H^p(L²(M)) is trivial if the total curvature of M is less than a positive constant depending only on n. Second, we show that the dimension of H^p(L²(M)) is finite if the total curvature of M is finite. The vanishing theorem is a generalized version of Gan-Zhu-Fang theorem and the finiteness theorem is an extension of Zhu-Fang theorem.

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