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Holomorphic curvature of complex Finsler submanifolds

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Science china Mathematics 2010年第7期53卷 263-276页

作者： zhong chunping school of mathematical sciences, xiamen university, xiamen 361005, china 1. school of mathematical sciences Xiamen University Xiamen 361005 China

Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F. Let M be a complex m-dimensional submanifold of M, which is endowed with the induced complex Finsler metric F. Let D be the complex Rund connection associated with (M, F). We prove that (a) the holomorphic curvature of the induced complex linear connection  on (M, F) and the holomorphic curvature of the intrinsic complex Rund connection ～* on (M, F) coincide; (b) the holomorphic curvature of ～* does not exceed the holomorphic curvature of D; (c) (M, F) is totally geodesic in (M, F) if and only if a suitable contraction of the second fundamental form B(·, ·) of (M, F) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (M, F).

关键词： Holomorphic curvature complex Finsler metric complex Finsler submanifold

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Hodge decomposition theorem on strongly Kahler Finsler manifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

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Science china Mathematics 2006年第11期 1696-1714页

作者： zhong chunping & zhong Tongde school of mathematical sciences, xiamen university, xiamen 361005, china Department of Mathematics, Zhejiang university, Hangzhou 310028, china

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (?)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M,F) is a compact strongly Kahler Finsler manifold, we define a (?)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in Kahler manifolds is still true in the more general compact strongly Kahler Finsler manifolds.

关键词： strongly Kahler Finsler Hodge decomposition theorem (?)-Laplacian.

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

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Science china Mathematics 2024年第8期67卷 1827-1864页

作者： Xiaoshu Ge chunping zhong school of mathematical sciences Xiamen UniversityXiamen 361005China

In this paper,a class of holomorphic invariant metrics is introduced on the irreducible classical domains of typesⅠ-Ⅳ,which are strongly pseudoconvex complex Finsler metrics in the strict sense of Abate and Patrizio(1994).These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found in the literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed to admit differential geometry *** are,however,not necessarily Hermitian quadratic as the Bergman *** metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of typesⅠ-Ⅳ,respectively,and they are all proved to be complete K?hler-Berwald *** enjoy very similar curvature properties as those of the Bergman metric on the irreducible classical domains,i.e.,their holomorphic sectional curvatures are bounded between two negative constants and their holomorphic bisectional curvatures are always nonpositive and bounded below by negative constants,*** the viewpoint of complex analysis,these metrics are analogs of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the sense of Chern(1996).

关键词： holomorphic invariant metric Kahler-Berwald metric irreducible classical domains

Horizontal ∂-Laplacian on complex Finsler manifolds

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Science china Mathematics 2005年第z1期48卷 377-391页

作者： zhong chunping & zhong Tongde school of Mathematics Science, xiamen university, xiamen 361005, china school of Mathematics Science Xiamen University Xiamen 361005 China

A horizontal (?)-Laplacian is defined on strongly pseudoconvex complex Finsler manifolds, first for functions and then for horizontal differential forms of type (p,q). The principal part of the (?)-Laplacian is computed in local coordinates. As an application, the (?)-Laplacian on strongly Kahler Finsler manifold is obtained explicitly in terms of the horizontal covariant derivatives of the Chern-Finsler conncetion.

关键词： complex Finsler manifold Kahler Finsler manifold ■-Laplacian horizontal.

The Koppelman-Leray formula on complex Finsler manifolds

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Science china Mathematics 2005年第6期48卷 847-863页

作者： QIU Chunhui & zhong Tongde school of mathematical sciences, xiamen university, xiamen 361005, china Academy of Mathematics and System sciences, Chinese Academy of sciences, Beijing 100080, china 1. school of mathematical sciences Xiamen University 361005 Xiamen China 2. Academy of Mathematics and System sciences Chinese Academy of Sciences 100080 Beijing China

By means of the invariant integral kernel (the Berndtsson kernel), the complex Finsler metric and the non-linear connection associated with the Chern-Finsler connection to research into the integral representation theory on complex Finsler manifolds, theKoppelman and Koppelman-Leray formulas are obtained, and the - -equations are solved.

关键词： complex Finsler manifold, Chern-Finsler connection, invariant integral kernel, Koppelman-Lerayformulas, - -equation.

STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

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Acta Mathematica Scientia 2010年第3期30卷 799-809页

作者：谭忠王焰金 school of mathematical sciences Xiamen University

In this article, we are concerned with the strong solutions for the incompress- ible fluid models of Korteweg type in a bounded domain Ω СR^3. We prove the existence and uniqueness of local strong solutions to the initial boundary value problem. We point out that in this article we allow the existence of initial vacuum provided initial data satisfy a compatibility condition.

关键词： Korteweg model strong solutions incompressible fluids Navier-Stokes equations vacuum

HIGHER ORDER SINGULAR INTEGRAL EQUATIONS ON COMPLEX HYPERSPHERE

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Acta Mathematica Scientia 2010年第5期30卷 1785-1792页

作者：陈吕萍钟同德 school of mathematical sciences Xiamen University

A theory of a class of higher order singular integral under the operator （L f）（u） =[u1 σf/σu1（u） - u1σf/σu1（u）＋ f（u）] is given. We transform the higher order singular integral to a usual Cauchy integral, extend the permutation formula of the higher order singular integral deduced by Qian and zhong in [4] to a general case, and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.

关键词： complex hypersphere Cauchy integral higher order singular integral Hadamard principal value permutation formula

CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

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Acta Mathematica Scientia 2011年第4期31卷 1449-1456页

作者：许建开谭忠 College of sciences Hunan Agriculture University school of mathematical sciences Xiamen University

In this paper, we consider the following integral system： u（x） = R n v q （y）｜ x y ｜ nα dy, v（x） = R n u p （y）｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes... 详细信息

In this paper, we consider the following integral system： u（x） = R n v q （y）｜ x y ｜ nα dy, v（x） = R n u p （y）｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of （0.1） are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].

关键词： Hardy-Littlewood-Sobolev inquality integral equation moving plane interpolation inequality radial symmetry

Some rigid results on doubly twisted product complex Finsler manifolds

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Science china Mathematics 2024年

作者： Hongchuan Xia Wei Xiao chunping zhong school of Mathematics and Statistics Xinyang Normal University school of mathematical sciences Xiamen University

Let(M1×（λ1,λ2）M2, F) be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds（M1, F1） and（M2, F2）. Denote by M the product manifold of M1and M2. In this paper, we first give the characterization for（M, F） to be of constant holomorphic curvature, and prove that（M, F） is of constant holomorphic curvature if and only if（M, F）,（M1, F1）, and（M2, F2） all have vanishing holomorphic curvature under the condition that ln λ1and ln λ2are both pluriharmonic ***, we obtain the classification for（M, F） to be a complex Landsberg manifold. Finally, we prove that a strongly convex doubly twisted product complex Finsler manifold（M, F） is projectively flat（resp. dually flat）if and only if λ1F1and λ2F2are both complex Minkowski metrics.

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