Showing papers on "Ricci flow published in 1994"

Metrics of negative Ricci curvature

Abstract: One of the most natural and important topics in Riemannian geometry is the relation between curvature and global structure of the underlying manifold. For instance, complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic. Furthermore, it seemed to be a natural principle that a (closed) manifold cannot carry two metrics of different signed curvatures, as it is a basic fact that this is true for sectional curvature. But it turned out to be wrong (much later and from a strongly analytic argument) for the scalar curvature S, since each manifold M', n > 3, admits a complete metric with S _-1 (cf. Aubin [A] and Bland, Kalka [BIK]). Hence the situation for Ricci curvature Ric, lying between sectional and scalar curvature, seemed to be quite delicate. Up to now, the most general results concerning Ric < 0 were proved by Gao, Yau [GY] and Brooks [Br] using Thurston's theory of hyperbolic threemanifolds, viz.: Each closed three-manifold admits a metric with Ric < 0. This is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities; Gao and Yau (resp. Brooks) smoothed these singularities to get a regular metric with Ric < 0. These methods extend to three-manifolds of finite type and certain hyperbolic orbifolds. In any case, the arguments rely on exploiting some extraordinary metric structures, whose existence is neither obvious nor conceptually related to the Ricci curvature problem. Indeed, the existence depends on the assumption that the manifold is three-dimensional and compact. Moreover this approach does not provide insight into the typical behaviour of metrics with Ric < 0 since one is led to very special metrics. In this article we approach negative Ricci curvature using a completely different and new concept (which will become even more significant in [L2]) as we deliberately produce Ric < 0. Actually we will prove the following results; in these notes Ric(g), resp. r(g), denotes the Ricci tensor, resp. curvature of a smooth metric g:

Manifolds of positive Ricci curvature with almost maximal volume

Abstract: 10. In this note we consider complete Riemannian manifolds with Ricci curvature bounded from below. The well-known theorems of Myers and Bishop imply that a manifold Mn with Ric > n 1 satisfies diam(Mn) < diam(Sn(l)), Vol(Mn) < Vol(Sn(l)). It follows from [Ch] that equality in either of these estimates can be achieved only if Mn is isometric to Sn (1) . The natural conjecture is that a manifold Mn with almost maximal diameter or volume must be a topological equivalent to Sn . With respect to diameter this is true only if Mn satisfies some additional assumptions; see [An, 0, GP, E]. With respect to volume however no extra restriction is necesary.

New examples of complete Ricci solitons

Abstract: The Ricci soliton condition reduces to a set of ODEs when one assumes that the metric is a doubly-warped product of a ray with a sphere and an Einstein manifold. If the Einstein manifold has positive Ricci curvature, we show there is a one-parameter family of solutions which give complete noncompact Ricci solitons. INTRODUCTION A Ricci soliton is a solution to the Ricci flow ag/Ot = -2Ric(g) such that the metric changes only by diffeomorphisms as time goes on; since the diffeomorphisms of the underlying manifold are symmetries of the evolution equation, it would be more accurate to call this a similarity solution for the Ricci flow. Soliton solutions are important to the study of the Ricci flow because they represent extremal cases for the Harnack estimate [H2] and may be limiting cases for the Ricci flow near singularities (cf. [A]). A Ricci soliton is generated by an initial metric g and a vector field V such that Svg = 2Ric(g); then V generates the diffeomorphisms. A gradient soliton is one where V is the gradient of some function h with respect to g; the corresponding condition is that the Hessian V2 h coincide with the Ricci tensor. Up to now, the known examples of complete Ricci solitons were the radially symmetric 'cigar' metric on R2 [Hi], the radially symmetric soliton on R3 discovered by Bryant (which easily generalizes to Rn), and the U(n)symmetric soliton on Cn disovered by Cao [C]. Let (Mn, da2) be a compact Einstein manifold with Einstein constant e > 0 . Let dO2 denote the standard metric of constant curvature + 1 on Sk, k > 1 . On Rk+1 x M with radial coordinate t > 0 consider the doubly-warped product metric (*) ds2 = dt2 + f(t)2d02 + g(t)2da2. For the metric to be smooth near t = 0 we require that f extend smoothly to Received by the editors December 2, 1992. 1991 Mathematics Subject Classification. Primary 53C25; Secondary 34C99.

A new approach to the Ricci flow on $S^2$

Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Ricci tensors of real hypersurfaces in a complex projective space

Abstract: This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector 4 is principal, the focal map has constant rank, and VyS = 0, where S is the Ricci tensor of the real hypersurface.

Negative ricci curvature and isometry group

Abstract: We show that for n-dimensional manifolds with Ricci curvature bounded between two negative constants the order of their isometry groups is uniformly bounded by the Ricci curvature bounds, the volume, and the injectivity radius. We also show that the degree of symmetry (seex2 for denition) is lower semicontinuous in the Gromov-Hausdor topology.

Rigidity and sphere theorem for manifolds with positive ricci curvature

Abstract: LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound.

On open three manifolds of quasi-positive Ricci curvature

Abstract: It is proved that an open three-manifold of Ricci curvature nonnegative and positive at one point is diffeomorphic to the three-dimensional Euclidean space

Ricci curvature and a criterion for simple-connectivity on the sphere

Abstract: From the recent work of Osgood and Stowe on the Schwarzian derivative for conformai maps between Riemannian manifolds we derive a sharp sufficient condition for a domain on the sphere to be simply-connected. We show further that a less restrictive form of the condition yields a uniform lower bound for the length of closed geodesies. Introduction Osgood and Stowe have recently defined a notion of Schwarzian derivative for conformai mappings of Riemannian manifolds which generalizes the classical operator for analytic functions in the plane [O-Sl]. As in complex analysis, where the Schwarzian derivative has been central as a means of characterizing conditions for global univalence, these authors establish in [0-S2] an injectivity criterion for conformai local diffeomorphisms y of a Riemannian «-manifold (M, g) to the standard sphere Sn . The univalence of y/ follows from a bound on the norm of the Schwarzian derivative by geometric quantities of M (Theorem 1.1). This result allows a unified approach to a vast class of injectivity theorems in the plane, as different criteria can be derived from it on a given domain just by changing the metric g conformally. Indeed, in [0-S2] the authors obtain as corollaries, with M the unit disc in the plane and g alternately the euclidean and hyperbolic metric, two classical conditions of Nehari. Most of the known criteria, including a recent injectivity result of Epstein [Ep], and some new conditions on the unit disc and simply-connected domains are derived in [Chi] from Theorem 1.1. We shall show in this paper that a local diffeomorphism ip as before satisfying a particular form of the criterion in [0-S2] forces the manifold M to be simply-connected (Theorem 2.1). Our main result, a sharp criterion for simpleconnectivity for domains in S" , will appear as a reformulation of Theorem 2.1 when using conformai invariance. This allows one to translate the existence of y/ to that of a conformai metric on a domain iicS" with the property that Received by the editors October 9, 1991 and, in revised form, January 8, 1993. 1991 Mathematics Subject Classification. Primary 53A30; Secondary 34C10.

Ricci curvature and holomorphic convexity in Kähler manifolds

Abstract: In this paper we show that if a smoothly bounded, relatively compact domain in a complex manifold admits a complete Kahler metric with certain bounds on its Ricci tensor, then the domain must be holomorphically convex. This gives an obstruction for the existence of a complete Kahler-Einstein metric on such domains.