The weighted Yamabe problem with boundary

2022/08/24 by Pak Tung Ho, Jinwoo Shin, Ho, Pak Tung +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.11310

openalex publication_date 2022/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a Yamabe-type flow \ (∂ g)/(∂ t) · amp;=(r^m_ϕ-R^m_ϕ)g
(∂ ϕ)/(∂ t) · amp;=(m)/(2)(R^m_ϕ-r^m_ϕ) . ~~ in M ~~ and ~~ H^m_ϕ=0 ~~ on ∂ M on a smooth metric measure space with boundary (M,g, v^mdV_g,v^mdA_g,m), where R^m_ϕ is the associated weighted scalar curvature, r^m_ϕ is the average of the weighted scalar curvature, and H^m_ϕ is the weighted mean curvature. We prove the long-time existence and convergence of this flow.

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• Convergence rate of the weighted Yamabe flow
• The Yamabe flow on asymptotically flat manifolds
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• Normalized Yamabe flow on manifolds with bounded geometry