The Calabi flow with rough initial data

2017/01/24 by Weiyong He, Y. Zeng, He, Weiyong +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1701.06943

openalex publication_date 2017/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that there exists a dimensional constant δ> 0 such that given any background Kähler metric ω, the Calabi flow with initial data u₀ satisfying ∂ ∂ u₀ ∈ L^∞ (M) and (1- δ)ωlt; ω_u0 lt; (1+δ)ω, admits a unique short time solution and it becomes smooth immediately, where ω_u0 : = ω+√(-1)∂ ∂ u₀. The existence time depends on initial data u₀ and the metric ω. As a corollary, we get that Calabi flow has short time existence for any initial data satisfying ∂ ∂ u₀ ∈ C⁰(M) and ω_u0 gt; 0, which should be interpreted as a "continuous Kähler metric". A main technical ingredient is Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time weighted Hölder norms.

Citations

• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics

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• Twisted Calabi functional and twisted Calabi flow

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