L_f^p harmonic 1-forms on complete non-compact smooth metric measure spaces

2020/05/13 by Jiuru Zhou, Peng Zhu, Zhou, Jiuru +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2005.06124

Abstract

This paper studies complete non-compact smooth metric measure space (Mⁿ,g,e^-fdv) with positive first spectrum λ₁(Δ_f) or satisfying a weighted Poincaré inequality with weight function ρ. We establish two splitting and vanishing theorems for L_f^p harmonic 1-forms under the assumption that m-Bakry-Émery Ricci curvature Ric_m,n≥ -aλ₁(Δ_f) or Ric_m,n≥ -aρ-b for particular constants a and b>0. These results are inspired by the work of Han-Lin and are L_f^p generalizations of previous works by Dung-Sung and Vieira for L² harmonic 1-forms.

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