Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space. (January 2017)

• Record Type: Journal Article
• Authors: Lin, Hezi
• Is Part Of: Nonlinear analysis. Volume 148(2017)
• Page Start: 126
• Page End: 137
• Publication Date: 2017-01
• Subjects: 53C21 -- 53C42; Complete submanifolds -- First eigenvalue -- Gap theorems -- 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.2016.09.015
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let M n be a complete non-compact submanifold in the hyperbolic space H n + m . We first give an estimate for the bottom of the spectral of the Laplace operator on M n, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L 2 harmonic forms in certain degrees if the total mean curvature of M n is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L 2 harmonic 1 -forms, we can further prove some one-end theorems.