RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

Asymptotic rank of spaces with bicombings

The question, under what geometric assumptions on a space X an n-quasiflat in X implies the existence of an n-flat therein, has been investigated for a long time. It was settled in the affirmative for Busemann spaces by Kleiner, and for manifolds of non-positive curvature it dates back to Anderson and Schroeder. We generalize the theorem of Kleiner to spaces with bicombings. This structure is a weak notion of non-positive curvature, not requiring the space to be uniquely geodesic. Beside a metric differentiation argument, we employ an elegant barycenter construction due to Es-Sahib and Heinich by means of which we define a Riemannian integral serving us in a sort of convolution operation.     

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.     

The structure of singular spaces of dimension 2

In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature. Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999     

Weil-Petersson metric in the moduli space of compact polarized Kähler-Einstein manifolds of zero first Chern class

We study the problem of computing the curvature of the Weil-Petersson metric of the moduli space of general compact polarized Kähler-Einstein manifolds of zero first Chern class. We use canonical lifting of vector fields from the moduli space to the total deformation space to obtain a formula for the curvature of the Weil-Petersson metric. From this formula we obtain negative bisectional curvature for certain directions. This formula also reprove and explain the recent result of Schumacher that the holomorphic sectional curvature of the Weil-Petersson metric for K3-surfaces and symplectic manifolds are negative.     

The Totally Geodesic Coisotropic Submanifolds in Kähler Manifolds

In this paper, we consider the coisotropic submanifolds in a Kähler manifold of nonnegative holomorphic curvature. We prove an intersection theorem for compact totally geodesic coisotropic submanifolds and discuss some topological obstructions for the existence of such submanifolds. Our results apply to Lagrangian submanifolds and real hypersurfaces since the class of coisotropic submanifolds includes them. As an application, we give a fixed-point theorem for compact Kähler manifolds with positive holomorphic curvature. Also, our results can be further extended to nearly Kähler manifolds.     

On a natural algebraic affine connection on the space of almost complex structures and the curvature of teichmüller space with respect to its Weil-Petersson metric

A formula for the sectional curvature of Teichmüller space with respect to the Weil-Petersson metric is derived in terms of the Laplace-Beltrami operator on functions. It will be shown that the sectional curvature as well as the holomorphic sectional curvature and Ricci curvature are negative. Bounds on the holomorphic and the Ricci curvature are given.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi–Yau threefold, conjecturally corresponding to approximations to the Weil–Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kähler cone is seen to be analogous to a slice of the Weil–Petersson metric near large complex structure limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil–Petersson metric has non-positive scalar curvature in some neighborhood of the large complex structure limit point.     

Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

New results on the proximal point algorithm in non-positive curvature metric spaces

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

Hyperkähler Syz Conjecture and Semipositive Line Bundles

Let M be a compact, holomorphic symplectic Kähler manifold, and L a non-trivial line bundle admitting a metric of semipositive curvature. We show that some power of L is effective. This result is related to the hyperkähler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.     

一类拟凸域的Bergman度量与Kobayashi度量的比较定理

本文对一类拟凸域Ｅ（ｍ，ｎ，Ｋ）给出其不变Ｋａｈｌｅｒ度量下的全纯截曲率的显表达式，并构造了Ｅ（ｍ，ｎ，Ｋ）的一个不变的完备的Ｋａｈｌｅｒ度量，使得它大于或等于Ｂｅｒｇｍａｎ度量，而且其全纯截曲率的上界是一个负常数，从而得到Ｅ（ｍ，ｎ，Ｋ）的Ｂｅｒｇｍａｎ度量和Ｋｏｂａｙａｓｈｉ度量的比较定理。     

Comparison theorem on Cartan-Hartogs domain of the first type

In this paper the holomorphic sectional curvature under invariant Kahler metrics on Cartan-Hartogs domain of the first type are given in explicit forms. In the meantime, we construct an invariant Kahler metric, which is not less than Bergman metric such that its holomorphic sectional curvature is bounded from above by a negative constant. Hence we obtain the comparison theorem for the Bergman metric and Kobayashi metric on Cartan-Hartogs domain of the first type.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2 × V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

Examples of Hermitian Manifolds with Pointwise Constant Holomorphic Sectional Curvature

We construct non-compact examples of Hermitian manifolds with pointwise constant holomorphic sectional curvature. Our examples are obtained by conformal change of the metric on an open set of the complex space form.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Schwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold

The authors prove the Schwarz lemma from a compact complex Finsler manifold to another complex Finsler manifold and any complete complex Finsler manifold with a non-positive holomorphic curvature obeying the Hartogs phenomenon.