Heat kernel estimate on weighted Riemannian manifolds under lower N-Ricci curvature bounds with ε-range and it's application

2025/05/25 by Wenqi Li, Li, Wen-Qi, Zhikai Zhang +1
Mathematics · #Analysis of PDEs (math.AP) #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.2505.19113

openalex publication_date 2025/05/25 · openalex created_date 2025/09/29 · openalex updated_date 2026/07/31

Abstract

In this paper, we establish a parabolic Harnack inequality for positive solutions of the ϕ-heat equation and prove Gaussian upper and lower bounds for the ϕ-heat kernel on weighted Riemannian manifolds under lower N-Ricci curvature bound with ε-range. Building on these results, we demonstrate: The L¹_ϕ-Liouville theorem for ϕ-subharmonic functions, L¹_ϕ-uniqueness property for solutions of the ϕ-heat equation and lower bounds for eigenvalues of the weighted Laplacian Δ_ϕ. Furthermore, leveraging the Gaussian upper bound of the weighted heat kernel, we construct a Li-Yau-type gradient estimate for the positive solution of weighted heat equation under a weighted L^p(μ)-norm constraint on |∇ϕ|².

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