L2 harmonic forms on submanifolds in a Hadamard manifold. (September 2015)

• Record Type: Journal Article
• Authors: Lin, Hezi
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 310
• Page End: 322
• Publication Date: 2015-09
• Subjects: 53C21 -- 53C42; Complete submanifolds -- Finite index -- L2-harmonic p-forms -- Ends; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.05.022
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In the present paper, we first show that an n − 1 n -super-stable complete minimal submanifold in R n + m admits no nontrivial L 2 harmonic 1 -forms, and has only one end, which generalized Cao–Shen–Zhu's result on stable minimal hypersurface in R n + 1 . For L 2 harmonic forms of higher order, we prove a vanishing and finiteness theorem under the assumptions on Schrödinger operators involving the squared norm of the traceless second fundamental form. Finally, if the potential function of the Schrödinger operator is a multiple of the squared norm of the second fundamental form, we can further show that the hypersurface has only one end or finitely many ends.