AbstractIn this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni,
An optimal gap theorem,
Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam,
Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature,
J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu,
Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds,
Duke Math. J. 165 2016, 15, 2899–2919].
In this note, we give a new proof of the nonnegativity of the scalar curvature under the condition of the nonnegative quadratic orthogonal bisectional curvature.
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.
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