We prove that the volume of a compact connected K\"ahler manifold with holomorphic sectional curvature at least 2 is bounded above by the volume of the Fubini-Study metric of constant holomorphic sectional curvature 2 on the complex projective space of the same dimension. Moreover equality holds if and only if the manifold is biholomorphically isometric to complex projective space. This answers a question posed by Xiong and Yang. Our approach also yields a different proof of Zhang's sharp volume estimate and Liu's rigidity theorem for compact K\"ahler manifolds with positive Ricci curvature. In fact, our main result states that the same sharp volume estimate holds under a new curvature positivity condition (mean RC curvature positivity), which is implied by both positive Ricci curvature and positive holomorphic sectional curvature. The definition of this condition was inspired by the work of Yang. The proofs in this paper are due to ChatGPT 5.6 Sol Pro, and the paper is merely an exposition of its output. The proofs has been verified by the authors and they take full responsibility for any errors.
An old conjecture in Hermitian geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature is K\"ahler when the constant is nonzero and Chern flat when the constant is zero. Yang--Zheng introduced the real bisectional curvature as a generalization of the holomorphic sectional curvatu...
We prove rigidity results for Hermitian metrics of constant Chern holomorphic sectional curvature and construct counterexamples to the flatness conjecture. On a compact complex manifold in Fujiki's class $\mathcal C$ of dimension $n\ge2$, every such metric with nonpositive curvature is K\"ahler; its universal cover is...
A long-standing conjecture in non-K\"ahler geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature should be K\"ahler when the constant is nonzero and Chern flat when the constant is zero. In this article, we study the corresponding problem for a Hermitian connection $\nabl...
We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive \(k^{\mathrm{th}}\)-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curva...
Mean curvature flow (MCF) is an effective tool for investigating the geometry and topology of submanifolds under suitable curvature pinching conditions. While most existing convergence results in complex projective spaces rely on pointwise curvature assumptions, much less is known about the case of integral curvature c...
Colding established a sharp gradient estimate for the Green function on manifolds with nonnegative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely...
Junyoung Kim, Jiewon Park· 0 citations
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