Let $\Omega\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\Omega$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be K\"ahler--Einstein. Consequently, extremality, constant scalar curvature, and the K\"ahler--Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using H\"ormander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and K\"ahler--Einstein throughout the locus where $K_\Omega(z,z)>0$.
We prove that every Gromov elliptic domain $\Omega$ in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood $V$ such that $\Omega\cap V$ becomes Stein after removing a set which is, locally in $\Omega\cap V$, the union of the polar sets of fini...
We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini...
C. Arezzo, A. Loi, G. Placini et al.· 1 citation· ⚡1
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
In this paper, we study some geometric properties of the disk bundles over the complete K\"ahler manifolds. When the base space is the complete K\"ahler-Einstein manifold, we compute the holomorphic sectional curvature, Riemannian sectional curvature of the complete K\"ahler-Einstein metric on the disk bundle. Moreover...
Yi-Hong Hao, Ming-Ming Chen, An Wang et al.· 0 citations
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
Let $n\geq3$ and $0\leq\tau\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindric...
Tian-Ci Luo, Yong Wei, Rong Zhou· 0 citations
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