The Chern-Ricci flow and holomorphic bisectional curvature

The Sasaki-ricci Flow and Compact Sasaki Manifolds of Positive Transverse Holomorphic Bisectional Curvature

We show that Perelman’s W functional on Kahler manifolds has a natural counterpart on Sasaki manifolds. We prove, using this functional, that Perelman’s results on Kahler-Ricci flow (the first Chern class is positive) can be generalized to Sasaki-Ricci flow, including the uniform bound on the diameter and the scalar curvature along the flow. We also show that positivity of transverse bisectiona...

The Holomorphic Bisectional Curvature of the Complex Finsler Spaces

The notion of holomorphic bisectional curvature for a complex Finsler space (M, F ) is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. By means of holomorphic curvature and holomorphic flag curvature of a complex Finsler space, a special approach is emloyed to obtain the characterizations of the holomorphic bisectional curvature. For the class of gen...

Characterization of Holomorphic Bisectional Curvature of GCR-Lightlike Submanifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature of GCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for a GCR-lightlike submanifold of an indefinite comple...

The Kähler - Ricci flow on Kähler manifolds with 2 traceless bisectional curvature operator

A Note on Compact Kähler-ricci Flow with Positive Bisectional Curvature

We show that for any solution gij̄(t) to the Kähler-Ricci flow with positive bisectional curvature Rīijj̄(t) > 0 on a compact Kähler manifold M , the bisectional curvature has a uniform positive lower bound Rīijj̄(t) > C > 0. As a consequence, gij̄(t) converges exponentially fast in C ∞ to an KählerEinstein metric with positive bisectional curvature as t → ∞, provided we assume the Futaki-invariant...