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     

Rigidity of conformally compact manifolds with the round sphere as the conformal infinity

In this paper, we prove that under a lower bound on the Ricci curvature and an assumption on the asymptotic behavior of the scalar curvature, a complete conformally compact manifold whose conformal boundary is the round sphere has to be the hyperbolic space. It generalizes similar previous results where stronger conditions on the Ricci curvature or restrictions on dimension are imposed.     

紧流形上椭圆微分算子预解式的一致L^p-L^q估计——献给陆善镇教授75华诞

假设n和m是两个正整数,P（x,D）是定义在维数为n的紧致无边流形M上的一般m阶椭圆自伴微分算子.在一定条件下,本文主要证明微分算子P（x,D）的预解式的一致L^p-L^q估计,其中n〉m≥2,（p,q）在Sobolev线上并满足1/p-1/q=m/n,p≤2（n＋1）/n＋3,q≥2（n＋1）/n-1.本文的一个核心引理是建立曲面Σx={ξ∈Tx^＊（M）：p（x,ξ）=1}上测度的Fourier变换衰减估计的具体表达式,并利用它来得到局部算子的一致L^p-L^q估计.     

The Huber theorem for non-compact conformally flat manifolds

It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Received: April 14, 2000; revised version: March 20, 2001     

Some compact conformally flat manifolds with non-negative scalar curvature

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.     

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Renormalizing curvature integrals on Poincaré-Einstein manifolds

After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well known and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.     

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we construct the high energy resolvent on general non-trapping asymptotically hyperbolic manifolds, using semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, together with the 0-calculus of Mazzeo-Melrose.

Our results generalize recent work of Melrose, Sá Barreto and Vasy, which applies to metrics close to the exact hyperbolic metric. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.     

Groups acting transitively on compact CR manifolds of hypersurface type

Let be a compact homogeneous manifold with acting effectively and with a -invariant CR structure of hypersurface type; then any maximal compact subgroup acts transitively on .

     

Islands at infinity on manifolds of asymptotically nonnegative curvature

We introduce an invariant which measures the R-eccentricity of a point in a complete Riemannian manifold M and show that it goes to zero when the point goes to infinity, if M has asymptotically nonnegative curvature. As a consequence we show that the isometry group is compact if M has asymptotically nonnegative curvature and a point with positive sectional curvature. Both authors were partially supported by CNPq of Brazil and the second author was also partially supported by FAPERJ of Brazil.     

Yamabe 流下完备Riemannian 流形在无穷远的性质

本文研究完备的局部共形平坦的Riemannian 流形Mⁿ. 证明了在Yamabe 流下, 流形在无穷远处曲率趋向于零的性质是随时间保持的. 作为应用, 可以得到这个流形的渐近体积比是一个常数.     

On compact symplectic manifolds with Lie group symmetries

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.

     

A note on 4-dimensional locally conformally flat Walker manifolds

A 4-dimensional Walker metric on a semi Riemanian manifold M, for the canonical metric with c = 0, have been investigated by M. Chaichi, E. García—Río and Y. Matsushita. The paper generalizes these notions to the case of constant c ≠ 0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found.     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.

     

The dual foliation in open manifolds with nonnegative sectional curvature

We study several properties of the Sharafutdinov dual foliation in open manifolds with nonnegative sectional curvature.

     

Rigidity for closed manifolds with positive curvature

Let M be an n-dimensional complete connected Riemannian manifold with sectional curvature sec(M) ≥ 1 and radius rad(M) > π/2. In this article, we show that M is isometric to a round n-sphere if for any x ∈ M, the first conjugate locus of x is a single point and if M contains a geodesic loop of length 2 · rad(M). We also show that the same conclusion is true if the conjugate value function at any point of M is a constant function. This work was done while the author was visiting MPI for Mathematics in Leipzig, Germany. The author is very grateful to MPI for Mathematics in Leipzig for its hospitality and CAPES.