The Koppelman-Leray formula on complex manifolds is obtained, and under suitable condition the continuous solution of partial derivative-equation on complex manifolds is obtained.
Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex mani...Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions h(z) of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.展开更多
A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman...A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman-Leray-Norguet formula without boundary integral of (r, s) differential forms, which is different from the classical one. The new weighted formula is especially suitable for the case of the local g-concave wedge with a non-smooth boundary, so one can avoid complex estimates of boundary integrals and the density of integral may be not defined on the boundary but only in the domain. Moreover, the weighted integral formulas have much freedom in applications such as in the interpolation of functions.展开更多
Let （M, F ） be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds （M 1 , F 1 ） and （M 2 , F 2 ）. In this paper, we obtain the relationship between the Chern Finsler conne...Let （M, F ） be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds （M 1 , F 1 ） and （M 2 , F 2 ）. In this paper, we obtain the relationship between the Chern Finsler connection coefficients Γ i ; k associated to F and the Chern Finsler connection coefficients Γ a ; c , Γα ; γ associated to F 1 , F 2 , respectively. As applications we prove that, if both （M 1 , F 1 ） and （M 2 , F 2 ） are strongly Ka¨hler Finsler （complex Berwald, or locally complex Minkowski, respectively） manifolds, so does （M, F ）. Furthermore, we prove that the holomorphic curvature K F = 0 if and only if K F1 = 0 and K F2 = 0.展开更多
In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 ...In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.展开更多
Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius tran...Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.展开更多
Studies are made of the cohomology of CR_ submanifolds and integrability of the distribution D of CR_submanifolds. When dim D⊥】1, the totally umbilical non-trival CR-submanifold i n nea r Kaehler manifold is totall...Studies are made of the cohomology of CR_ submanifolds and integrability of the distribution D of CR_submanifolds. When dim D⊥】1, the totally umbilical non-trival CR-submanifold i n nea r Kaehler manifold is totally geodesic. In the end, we get:If is n ear Kaehler manifold with B】0, then is not permitt ed to have fixed foliate non-trival CR-submanifold.展开更多
In this paper, some construction theorems of pluriharmonic maps into complex Grassmann manifolds axe obtained. By these, there exists a characterization of strongly isotropic pluriharmonic maps.
In this note we show that on any compact subdomain of a K?hler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer ...In this note we show that on any compact subdomain of a K?hler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K?hler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.展开更多
We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the g...We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.展开更多
Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities. In this work we propose a reliabl...Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities. In this work we propose a reliable method for constructing complex networks from chaotic time series. We first estimate the covariance matrices, then a geodesic-based distance between the covariance matrices is introduced. Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance, respectively. The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.展开更多
In this paper,we prove that for certain fiber bundles there is a k-Futaki-Ono conformally Kahler metric related to a metric in any given Kahler class for any k≥2.
Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard to...Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.展开更多
Let(M1×(λ_(1),λ_(2))M_(2),F)be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds(M_(1),F_(1))and(M_(2),F_(2)).Denote by M the product manifold of M1and M_(...Let(M1×(λ_(1),λ_(2))M_(2),F)be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds(M_(1),F_(1))and(M_(2),F_(2)).Denote by M the product manifold of M1and M_(2).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),(M_(1),F_(1)),and(M_(2),F_(2))all have vanishing holomorphic curvature under the condition that lnλ_(1)and lnλ_(2)are both pluriharmonic functions.Secondly,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 ifλ_(1)F1andλ_(2)F_(2)are both complex Minkowski metrics.展开更多
We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through developm...We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.展开更多
In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian cu...In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G（2, n ＋ 1） by making use of Veronese sequence.展开更多
In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strateg...In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.展开更多
文摘The Koppelman-Leray formula on complex manifolds is obtained, and under suitable condition the continuous solution of partial derivative-equation on complex manifolds is obtained.
文摘Analytic atlases on <img src="Edit_948e45b7-cbef-425e-bb79-28648b859994.png" width="23" height="22" alt="" /> can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing <img src="Edit_7e5721ee-bb7f-4224-bd52-7d4641ac1c73.png" width="23" height="22" alt="" /> with the two elements group of a fixed point free antianalytic involution of <img src="Edit_ddfdac64-b296-48c5-9bb2-932eb3d76008.png" width="23" height="22" alt="" />. Involutions h(z) of this kind are obtained linearly by composing special M<span style="white-space:nowrap;"><span style="white-space:nowrap;">öbius transformations of the planes with the mapping <img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="89" height="24" alt="" /><img src="Edit_4cda269a-9399-41ae-a5da-0c9d18a419ab.png" width="85" height="22" alt="" />. A convenient partition of <img src="Edit_9e899708-41b0-4351-a12b-cc6efb5b1581.png" width="23" height="22" alt="" /> is performed which helps defining an internal operation on <img src="Edit_7cd42987-68f8-4e4c-9382-cbc68b56377e.png" width="49" height="26" alt="" /> and finally actions of the previously defined Lie groups on the non orientable manifold <img src="Edit_5740b48c-f9ea-438d-a87d-8cdc1f83662b.png" width="49" height="25" alt="" /> are displayed.
基金National Natural Science Foundation and Mathematical "Tian Yuan" Foundation of China (10271097 and TY10126033)
文摘A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman-Leray-Norguet formula without boundary integral of (r, s) differential forms, which is different from the classical one. The new weighted formula is especially suitable for the case of the local g-concave wedge with a non-smooth boundary, so one can avoid complex estimates of boundary integrals and the density of integral may be not defined on the boundary but only in the domain. Moreover, the weighted integral formulas have much freedom in applications such as in the interpolation of functions.
基金supported by Program for New Century Excellent Talents in Fujian Provincial Universitythe Natural Science Foundation of China (10971170 10601040)
文摘Let （M, F ） be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds （M 1 , F 1 ） and （M 2 , F 2 ）. In this paper, we obtain the relationship between the Chern Finsler connection coefficients Γ i ; k associated to F and the Chern Finsler connection coefficients Γ a ; c , Γα ; γ associated to F 1 , F 2 , respectively. As applications we prove that, if both （M 1 , F 1 ） and （M 2 , F 2 ） are strongly Ka¨hler Finsler （complex Berwald, or locally complex Minkowski, respectively） manifolds, so does （M, F ）. Furthermore, we prove that the holomorphic curvature K F = 0 if and only if K F1 = 0 and K F2 = 0.
基金Supported by the National Natural Science Foundation of China (10531090)Knowledge Innovation Funds of CAS (KJCX3-SYW-S03)
文摘In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.
基金supported by the National Science Foundationsupported by a collaboration grant from the Simons Foundation(Grant No.523341)PSC-CUNY awards and a grant from NSFC(Grant No.11571122)。
文摘Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.
文摘Studies are made of the cohomology of CR_ submanifolds and integrability of the distribution D of CR_submanifolds. When dim D⊥】1, the totally umbilical non-trival CR-submanifold i n nea r Kaehler manifold is totally geodesic. In the end, we get:If is n ear Kaehler manifold with B】0, then is not permitt ed to have fixed foliate non-trival CR-submanifold.
基金Research supported by National Nature Science Foundation of China(10171012),Tian Yuan Foundation 10226001 and Foundation of Southeast University
文摘In this paper, some construction theorems of pluriharmonic maps into complex Grassmann manifolds axe obtained. By these, there exists a characterization of strongly isotropic pluriharmonic maps.
基金partially supported by ERC Consolidator Grant IPFLOW(Grant No.725967)supported by the Academy of Finland(Finnish Centre of Excellence in Inverse Problems Research(Grant Nos.284715 and 309963))+2 种基金by the European Research Council under FP7/2007-2013 ERC StG(Grant No.307023)Horizon 2020 ERC CoG(Grant No.770924)partially supported by Australian Research Council(Grant Nos.DP190103302 and DP190103451)
文摘In this note we show that on any compact subdomain of a K?hler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K?hler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.
文摘We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.
基金Supported by the National Natural Science Foundation of China under Grant No 61362024
文摘Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities. In this work we propose a reliable method for constructing complex networks from chaotic time series. We first estimate the covariance matrices, then a geodesic-based distance between the covariance matrices is introduced. Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance, respectively. The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.
基金supported by the Nature Science Foundation of China(12171140).
文摘In this paper,we prove that for certain fiber bundles there is a k-Futaki-Ono conformally Kahler metric related to a metric in any given Kahler class for any k≥2.
基金This paper is a talk on the held in Nanjing, P. R. China, July, 2004.
文摘Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.
基金supported by National Natural Science Foundation of China(Grant Nos.12071386 and 11701494)the Nanhu Scholars Program for Young Scholars of Xinyang Normal University。
文摘Let(M1×(λ_(1),λ_(2))M_(2),F)be a doubly twisted product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds(M_(1),F_(1))and(M_(2),F_(2)).Denote by M the product manifold of M1and M_(2).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),(M_(1),F_(1)),and(M_(2),F_(2))all have vanishing holomorphic curvature under the condition that lnλ_(1)and lnλ_(2)are both pluriharmonic functions.Secondly,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 ifλ_(1)F1andλ_(2)F_(2)are both complex Minkowski metrics.
文摘We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.
基金supported by the NSFC (11071248, 11071249)supported by the Fundamental Research Funds for the Central Universities(USTC)
文摘In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G（2, n ＋ 1） by making use of Veronese sequence.
文摘In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.
