非正曲率流形及其子流形上有界区域的特征值

本文研究了完备单连通具有非正曲率黎曼流形及其子流形上有界区域的特征值问题.利用广义Hessian比较定理,获得了局部特征值的下界估计式,将McKean[2]的定理在局部上推广到了非正曲率的情形.     

Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kähler manifolds and a sharp complex Hessian comparison on nearly Kähler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.     

An Alexandroff–Bakelman–Pucci estimate on Riemannian manifolds

In Cabré (1997) [2], Cabré established an Alexandroff–Bakelman–Pucci (ABP) estimate on Riemannian manifolds with non-negative sectional curvatures and applied it to establish the Krylov–Safonov Harnack inequality on manifolds with non-negative sectional curvatures. In the present paper, we generalize the results of [2]. We obtain an ABP estimate on manifolds with Ricci curvatures bounded from below and apply this estimate to prove the Krylov–Safonov Harnack inequality on manifolds with sectional curvatures bounded from below. We also use this ABP estimate to study Minkowski-type inequalities.     

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

On almost constant-type manifolds

We introduce the class of almost constant-type manifolds and prove some theorems concerning Ricci tensors, scalar curvatures, bisectional curvatures and curvature identities. The above class is also studied in relation to other known classes of almost hermitian manifolds.To Adriano Barlotti with friendship and esteemThis work has been partially supported by a contribution of Ministero Ricerca Scientifica e Tecnologica.     

Riemannian Manifold in Which the Skew-Symmetric Curvature Operator has Pointwise Constant Eigenvalues

We study manifolds where the natural skew-symmetric curvature operator has pointwise constant eigenvalues. We give a local classification (up to isometry) of such manifolds in dimension 4. In dimension 3, we describe such manifolds up to a classification of three - dimensional Riemannian manifolds with principal Ricci curvatures r₁ = r₂ = 0, r₃- arbitrary. We give examples of such manifolds in all dimensions which do not have constant sectional curvature; these manifolds are not pointwise Osserman manifolds in general.     

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17-28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R³, having the prescribed principal Ricci curvatures.     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

Higher order Hessian structures on manifolds

In this paper we define nth order Hessian structures on manifolds and study them. In particular, whenn = 3, we make a detailed study and establish a one-to-one correspondence betweenthird-order Hessian structures and acertain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.     

On the mean curvatures sharp estimates of hypersurfaces

Sharp estimates for the mean curvatures of hypersurfaces in Riemannian manifolds are known from the works of Jorge-Xavier ^[3], Markvorsen ^[6] and Vlachos ^[11]. We first give a simplified proof of these estimates. This proof shows that a similar original result holds for hypersurfaces in Einstein manifolds which are warped product of by Ricci-flat manifolds.     

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.The research of the Y. L. Xin was partially supported by NSFC of and SFECC.     

Lorentzian Submanifolds in Lorentzian Space Form with the Same Constant Curvatures

We study nondegenerate isometric immersions of Lorentzian manifolds of constant sectional curvatures into Lorentzian space form of the same constant sectional curvatures which have flat normal bundles. We also give a method to produce such immersions using the Lorentzian Grassmannian systems.     

Remark about scalar curvature and Riemannian submersions

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning scalar curvatures of smooth limit spaces arising in bounded curvature collapses.

     

Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime

We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett  29 .     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

Some compactness theorems via m-Bakry–Émery and m-modified Ricci curvatures with negative m

Stimulated by S. Ohta and W. Wylie, we establish some compactness theorems for complete Riemannian manifolds via m-Bakry–Émery and m-modified Ricci curvatures with negative m. Our results may be considered as generalizations of the classical compactness theorems via Ricci curvature due to S.B. Myers, W. Ambrose, G.J. Galloway, and J. Cheeger, M. Gromov, and M. Taylor, and relax some previous compactness criteria for complete Riemannian manifolds via m-Bakry–Émery and m-modified Ricci curvatures obtained when m is a positive constant or infinity.     

Regularity of Weak Solutions to a Class of Complex Hessian Equations on Kähler Manifolds

We prove the smoothness of weak solutions to a class of complex Hessian equations on closed Kähler manifolds, by use of the smoothing property of the corresponding gradient flow.     

Riemannian Newton method for positive bounded Hessian functions

In this work we study Newton type method for functions on Riemannian manifolds whose Hessian satisfies a double inequality. The main results refer to global convergence and convergence rate estimates.     

双扭曲积Hermitian流形

主要研究双扭曲积Hermitian流形的各种曲率,给出了紧致非平凡的双扭曲积Hermitian流形具有常全纯截面曲率的充要条件,得到了一种构造满足第一或第二爱因斯坦条件的Hermitian流形的有效方法.