Sharp differential estimates of Li‐Yau‐Hamilton type for positive (p, p) forms on Kähler manifolds

Abstract

In this paper we study the heat equation (of Hodge Laplacian) deformation of .p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a .p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack-type estimates for both the Kähler-Ricci flow and the Ricci flow.

Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds

“…We can also derive this from the classical Kodaira-Bochner formula for (1, 1)forms. (See for example [12] as well as Lemma 2.1 of [14].) This is based on the following observation: Let η denote the (1, 1)-form f * ω h with ω h being the Kähler form of N .…”

Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature

“…The approach of [12] toward Theorem 1.1 is via the asymptotic behavior of the optimal solution obtained by evolving a (1, 1)-form with the initial data being the Ricci form through the heat flow of the Hodge-Laplacian operator. The key component of the proof is the monotonicity obtained in [11] (see also [13]), which makes the use of the nonnegativity of the bisectional curvature crucially. On the other hand, in [17], the authors proved that the method of deforming a (1, 1)-form via the Hodge-Laplacian heat equation and studying the asymptotic behavior of the solution can be applied to solve the Poincaré-Lelong equation and obtain an optimal solution for it.…”

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

“…We can also derive this from the classical Kodaira-Bochner formula for .1; 1/-forms. (See, for example,[21] as well as lemma 2.1 of[26].) This is based on the following observation: Let denote the .1; 1/-form f £ !…”

“…h being the Kähler form of N . Then k@f k 2 is nothing but (following the notation of[26] with being the contraction using the Kähler metric ! g ).…”