On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

An Obata-type theorem on a three-dimensional CR manifold

We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three dimensional unit sphere.

Kohn-Rossi上同调与Bochner技巧

Kohn-Rossi cohomology and the Bochner technique

本文在具有非负曲率算子的紧致严格拟凸伪厄米特流形上证明了Betti数的消失定理，其核心方法是将Bochner技巧推广应用于CR流形。

Holomorphic sectional curvature of complex Finsler manifolds

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.

完备复Finsler度量与Kobayashi度量的一致等价性

Complete complex Finsler metrics and uniform equivalence of the Kobayashi metric

本文通过研究Bergman曲率有界性和Kähler流形曲率对称性，结合Schwarz引理，证明了有界强拟凸域或有界凸域上存在全纯截面曲率被负常数上界控制的完备强拟凸复Finsler度量，并进一步建立了有界强凸域上Kobayashi度量与Carathéodory度量的一致等价性。

Isometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $\mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.

The holomorphic sectional curvature and "convex" real hypersurfaces in Kähler manifolds

We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense o…

The k-Yamabe problem on CR manifolds

We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the …

A Schwarz lemma for weakly Kähler-Finsler manifolds

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseud…

Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hatθ$ so that $\hatθ$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hatθ)$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $…

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Suppose that $M$ is a Kähler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorp…

Bergman-Calabi diastasis and Kähler metric of constant holomorphic sectional curvature

We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation…

负向夹紧的全纯截面曲率的完备Kähler度量的不存在性

Non-existence of complete Kähler metric of negatively pinched holomorphic sectional curvature

本文证明了在有界弱伪凸域上，若Kähler度量满足特定曲率与几何条件，则其完备性决定了负向夹紧全纯截面曲率完备Kähler度量及完备Kähler-Einstein度量的存在性二分法，揭示了此类度量的不存在性充分条件。

Kähler 流形的 Wong-Rosay 型定理

The Wong-Rosay type theorem for Kähler manifolds

本文将 Wong-Rosay 定理从 $\mathbb{C}^n$ 的域推广至具有负截面曲率的单连通完备 Kähler 流形，并展示了具有全纯非不变度量流形的 Wong-Rosay 型定理。

完备非紧Kähler流形上经由Bergman核的不变度量等价性

Equivalence of Invariant metrics via Bergman kernel on complete noncompact Kähler manifolds

本文在完备非紧Kähler流形上，仅利用Bergman核与Bergman度量n次外积之比有界这一条件，证明了Bergman度量与完备Kähler–Einstein度量的等价性，并通过双参数拟凸域族验证了方法的有效性。