Vanishing theorems for f-harmonic forms on smooth metric measure spaces

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 L2f-harmonic forms on concrete geometric models. In the second...

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Bibliographic Details
Published inNonlinear analysis Vol. 162; pp. 113 - 127
Main Authors Han, Yingbo, Lin, Hezi
Format Journal Article
LanguageEnglish
Published Elmsford Elsevier Ltd 01.10.2017
Elsevier BV
Subjects
[formula omitted]-harmonic forms
Curvature
Smooth metric measure spaces
Space
Studies
Theorems
Vanishing theorems
53C20
f-harmonic forms
Smooth metric measure spaces
Vanishing theorems
53C21
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Summary: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 L2f-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 Lqf-harmonic 1-forms.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2017.06.012