Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact Kähler manifolds

2025/07/13 by Fusheng Deng, Gang Huang, Deng, Fusheng +3
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2507.09486

Abstract

The main goal of this paper is to generalize the Sobolev-type inequalities given by Guo-Phong-Song-Sturm and Guedj-Tô from the case of functions to the framework of twisted differential forms. To this end, we establish certain estimates of heat kernels for differential forms with values in holomorphic vector bundles over compact Kähler manifolds. As applications of these estimates, we also prove a vanishing theorem and give certain L^q,p-estimates for the ∂-operator on twisted differential forms.

Citations

• Some Sobolev-type inequalities for twisted differential forms on real and complex manifolds
• Diameter estimates in Kähler geometry II: removing the small degeneracy assumption
• Green's functions and complex Monge-Ampère equations
• Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms
• Gradient and Eigenvalue Estimates on the canonical bundle of Kähler manifolds
• Gradient estimate for eigenforms of Hodge Laplacian

Related

• Some Sobolev-type inequalities for twisted differential forms on real and complex manifolds
• Bergman kernels and eigenvalue estimate of ∂-laplacian
• Heat kernel asymptotics for Laplace type operators and matrix KdV hierarchy
• Gradient and Eigenvalue Estimates on the canonical bundle of Kähler manifolds
• L p -estimates for the heat semigroup on differential forms, and related\n problems