负曲率流形上给定数量曲率的共形形变

本文研究具强负曲率Cartan－Hadamard流形M~n（n≥3）上给定数量曲率函数S的共形形变问题．利用上下解方法,并通过精心构造上解,我们获得了当完备的共形形变度量存在时,函数S在无穷远附近的最佳渐近性态．在较一般情况下,我们还给出了共形数量曲率方程解的渐近估计．     

CH上的方程△u－hu＋fu^p＝０与黎曼度量的共形形变

设M是一个度量g的完备非紧非正曲率单连通黎曼流形，k是它的数曲率，K是M上的光滑函数．作者给出了M上以K作为数曲率且共形于g的度量的存在性条件，并给出了方程△u－hu＋fu^p＝０的一些正解存在性结果．     

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Let (M,g) be a complete simply-connected Riemannian manifold with nonpositive curvature, k its scalar curvature, and K a smooth function on M. We obtain a nonexistence result of complete metrics on M conformal to g and with K as their scalar curvature.     

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.     

An obstruction to the conformal compactification of Riemannian manifolds

In this paper, we study noncompact complete Riemannian -manifolds with which are not pointwise conformal to subdomains of any compact Riemannian -manifold. For this, we compare the Sobolev Quotient at infinity of a noncompact complete Riemannian manifold with that of the singular set in a compact Riemannian manifold using the method for the Yamabe problem.

     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

Finsler metrics with special Riemannian curvature properties

The Weyl curvature is one of the fundamental quantities in Finsler geometry because it is a projective invariant. By determining the Weyl curvature of a class of Finsler metrics, we find a lot of Finsler metrics of quadratic Weyl curvature which are non-trivial in the sense that they are not of quadratic Riemann curvature.     

On the curvature conjecture of Hua Loo-keng

It is proved that both the holomorphic sectional and the bisectional curvatures of the conformal Bergman metric ds21 = K2(z,)2log K(z, )/zαβdzαdβ are always negative, where K(z,) is the Bergman kernel of a bounded domain Din Cn . As a subsequent result, the Weyl tensor for a Hermitian manifold is obtained.     

Conformal scalar curvature equations on complete manifolds

This note contains considerations on the existence and non-existence problem of conformal scalar curvature equations on some complete manifolds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radius. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conformally compact manifolds. We obtain non-existence results on both types of conditions. Then we study in more details the existence problem on manifolds with cones, manifolds with cusps and conformally flat manifolds of bounded positive scalar curvature.     

Rigidity of Riemannian manifolds with positive scalar curvature

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the \(L^{\frac{n}{2}}\)-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties

In this paper, we study Finsler metrics of scalar flag curvature. We find that a non-Riemannian quantity is closely related to the flag curvature. We show that the flag curvature is weakly isotropic if and only if this non-Riemannian quantity takes a special form. This will lead to a better understanding on Finsler metrics of scalar flag curvature.      

Iteration-complexity of the subgradient method on Riemannian manifolds with lower bounded curvature

Abstract

The subgradient method for convex optimization problems on complete Riemannian manifolds with lower bounded sectional curvature is analysed in this paper. Iteration-complexity bounds of the subgradient method with exogenous step-size and Polyak's step-size are stablished, completing and improving recent results on the subject.     

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S ⁿ is a group.     

Negative exterior curvature foliations on compact Riemannian manifolds

The topological structure of compact Riemannian manifolds that admit hyperbolic foliations is studied. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 673–676, November, 1997. Translated by S. S. Anisov     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].