Locally Conformal K¨ahler and Hermitian Yang-Mills Metrics

The author shows that if a locally conformal K¨ahler metric is Hermitian YangMills with respect to itself with Einstein constant c ≤ 0, then it is a K¨ahler-Einstein metric.In the case of c > 0, some identities on torsions and an inequality on the second Chern number are derived.     

Curvature identities on almost Hermitian manifolds and applications

We systematically derive the Bianchi identities for the canonical connection on an almost Hermitian manifold.Moreover,we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection.As applications of the curvature identities,we obtain some results about the integrability of quasi K¨ahler manifolds and nearly K¨ahler manifolds.     

Holomorphic Lefschetz Fixed Point Formula for Non-compact Khler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact "hyperbolic" Khler manifolds (e.g. Khler hyperbolic manifolds, bounded domains of holomorphy) by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.     

Holomorphic Lefschetz Fixed Point Formula for Non-compact K¨ahler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kǎihler manifolds （e.g. Kǎihler hyperbolic manifolds, bounded domains of holomorphy） by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.     

本刊英文版2021年64卷第6期(1109–1330)摘要（英文）

正Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 Yuguang Shi Jintian Zhu Abstract By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time,     

On the Extension of Smale-Birkhoff Homoclinic Theorem

In this paper the S-B homoclinic theorem is extended to the following situation, If the stable and the unstable manifolds of some zero-dimensional hyperbolic basic sets for f∈ Diff (M, M) , dim M≥2, form a "transversal" circle, then there exists a "bigger" zero-dimensional hyperbolic basic set of f in M which includes all those original ones together with the points of intersection of their stable and unstable manifolds.     

Hyperbolic Khler-Ricci flow

In this paper, the author has considered the hyperbolic Khler-Ricci flow introduced by Kong and Liu, that is, the hyperbolic version of the famous Khler-Ricci flow. The author has explained the derivation of the equation and calculated the evolutions of various quantities associated with the equation including the curvatures. Particularly on Calabi-Yau manifolds, the equation can be simplifled to a scalar hyperbolic Monge-Ampère equation which is the hyperbolic version of the corresponding one in Khler-Ricci flow.     

Parallel Translation on K¨ahler Manifolds

In this paper, the author establishs a real-valued function on K¨ahler manifolds by holomorphic sectional curvature under parallel translation. The author proves if such functions are equal for two simply-connected, complete K¨ahler manifolds, then they are holomorphically isometric.     

VANISHING THEOREMS FOR ACH KAHLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincare inequality case and establish a vanishing theorem provided that the weighted function p is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincare inequality.     

Harmonic maps between compact Hermitian manifolds

In this paper, we generalize the Bochner-Kodaira formulas to the case of Hermitian com- plex (possibly non-holomorphic) vector bundles over compact Hermitian (possibly non-K¨ahler) mani- folds. As applications, we get the complex analyticity of harmonic maps between compact Hermitian manifolds.     

VANISHING THEOREMS FOR ACH KHLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Khler manifolds which are asymptotically complex hyperbolic near infinity . The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighte...     

A geometric heat flow for vector fields

We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).     

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.     

RESEARCH ANNOUNCEMENTS

For many years, the existence of chaotic behavior in dynamical systems has received much attention. Theoretical and experimental methods to show the existence of chaos have been well developed.It is well-known that if the stable and unstable manifolds of a hyperbolic periodic orbit     

THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T ′M : two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with K¨ahler spaces, in the two-dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the K¨ahler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0.Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact K¨ahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C_1~n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that,under bounded curvature conditions, C_1~n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on K¨ahler manifolds with minimal volume growth.     

L2-harmonic 1-forms on Complete Manifolds

We study the global behavior of complete minimal δ-stable hypersurfaces in R~(n+1) by using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ (n-1)~2/n~2)hypersurface in R~(n+1) has only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author's theorems on hypersurfaces and quaternionic K¨ahler manifolds.     

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone‘s for the three dimensional non-Hamiltonian systems.     

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper,we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into K¨ahler manifold.Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V=JMδJM satisfies some decay conditions,we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps.These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.     

Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the(1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics.More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are K¨ahler Calabi–Yau surfaces and Hopf surfaces.