Vanishing theorems for f-harmonic forms on smooth metric measure spaces. (October 2017)

• Record Type: Journal Article
• Authors: Han, Yingbo; Lin, Hezi
• Is Part Of: Nonlinear analysis. Volume 162(2017)
• Page Start: 113
• Page End: 127
• Publication Date: 2017-10
• Subjects: 53C21 -- 53C20; Vanishing theorems -- f-harmonic forms -- Smooth metric measure spaces; 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.2017.06.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we first establish a monotonicity formula for vector bundle-valued f -harmonic p -forms on a smooth metric measure space provided 〈 ∇ f, ∇ r 〉 is less that an explicit constant. As applications, we get some vanishing theorems for L 2 f -harmonic forms on concrete geometric models. In the second part, for a metric measure space with nonnegative ∞ -Bakry–Émery–Ricci curvature and with moderate volume growth, we prove that any bounded f -harmonic 1 -form must be parallel. Moreover, some vanishing theorems under nonnegative m -Bakry–Émery–Ricci curvature assumption are also proved. Finally, we consider smooth metric measure spaces with weighted Poincaré inequality and show some vanishing theorems for L q f -harmonic 1 -forms.