arXiv subjects

Xianchao Zhou

Publications and source records attributed to Xianchao Zhou.

6 recordsLinked to original sources

arXiv2025

On Kodaira dimension and scalar curvature in almost Hermitian geometry

In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or $κ(M, J)=0$, in which case $(M,J,g)$ is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero.

math.DG↗

arXiv2023

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

In this paper we consider the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds. By deducing the expression of the Gauduchon scalar curvature under the conformal variation, the problem is reduced to solve a semi-linear partial differential equation with exponential nonlinearity. Using super and sub-solution method, we show that the existence of the solution to this semi-linear equation depends on the sign of a constant associated to Gauduchon degree. When the sign is negative, we give both necessary and sufficient conditions that a prescribed function is the Gauduchon scalar curvature of a conformal Hermitian metric. Besides, this paper recovers Chern Yamabe problem, prescribed Chern Yamabe problem and Bismut Yamabe problem.

math.DG↗

arXiv2020

Totally Real Flat Minimal Surfaces in Quaternionic Projective Spaces

In this paper, we study totally real minimal surfaces in the quaternionic projective space $\mathbb{H}P^n$. We prove that the linearly full totally real flat minimal surfaces of isotropy order $n$ in $\mathbb{H}P^n$ are two surfaces in $\mathbb{C}P^n$, one of which is the Clifford solution, up to symplectic congruence.

math.DG↗

arXiv2019

Scalar curvatures in almost Hermitian geometry and some applications

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we get some inequalities of various total scalar curvatures and some characterization results of the Kähler metric, balanced metric, locally conformally Kähler metric and the k-Gauduchon metric. As corollaries, we show some results related to a problem given by Lejmi-Upmeier \cite{LeU} and a conjecture given by Angella-Otal-Ugarte-Villacampa \cite{AOUV}.

math.DG↗

arXiv2018

Twistor geometry of Hermitian surfaces induced by canonical connections

In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relations between the natural geometry of the twistor spaces and the geometry of Hermitian surfaces.

math.DG↗

arXiv2014

Twistor geometry of Riemannian 4-manifolds by moving frames

In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary connection on the vertical tangent bundle over the twistor space $Z$, we can recover J. Fine and D. Panov's result on the condition of the first Chern form being symplectic and P. Gauduchon's result on the condition of the first Chern form being a (1,1)-form respectively, by using the method of moving frames.

math.DG↗