Convergence of the Weighted Yamabe Flow

2021/11/10 by Zetian Yan, Yan, Zetian
Mathematics · #47J35 #53A30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Primary: 35G25. Secondary: 35K90

paper · pdf · doi:10.48550/arxiv.2111.05945

openalex publication_date 2021/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the weighted Yamabe flow (∂ g)/(∂ t)=(r^m_ϕ-R^m_ϕ)g, (∂ ϕ)/(∂ t)=(m)/(2)(R^m_ϕ-r^m_ϕ) on a smooth metric measure space (Mⁿ, g, e^-ϕ\rm dvol_g, m), where R^m_ϕ denotes the associated weighted scalar curvature, and r^m_ϕ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension n satisfies n\geqslant 3.

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