Showing papers on "Ricci flow published in 1998"

Recent Developments on the Ricci Flow

Abstract: This article reports recent developments of the research on Hamilton's Ricci flow and its applications.

The Ricci Curvature

Abstract: In this chapter we deal with problems concerning Ricci Curvature mainly: Prescribing the Ricci curvature Ricci curvature with a given sign Existence of Einstein metrics.

Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

Abstract: where dA -F, on the space of closed curves on the manifold Mn. Here A is a 1-form (i.e., F is an exact 2-form). This functional is a natural generalization of the usual functional of length, and its closed extremals correspond to periodic trajectories of the motion of particles on the Riemannian manifold Mn when the kinetic energy is defined by the metric tensor and the form F defines a magnetic field. Also this functional corresponds to the periodic orbits for other problems of classical mechanics and mathematical physics, as it was shown in [N2], [N3], [NS]. When the Lagrangian function

A Rigidity Theorem for Ricci Flat Metrics

Abstract: A rigidity result of the complete n-dimensional spin Ricci flat manifolds admitting a certain smooth S1 action is proved, provided that the action has fixed points and the metric is asymptotically flat. Such manifolds are isometric to the n-dimensional Riemannian Schwarzschild metric.

Surgery on ricci positive manifolds

Abstract: We consider performing surgery on Riemannian manifolds with positive Ricci curvature. We find conditions under which the resulting manifold also admits a positive Ricci curvature metric. These conditions involve dimension and the form taken by the metric in a neighbourhood of the surgery.

On 3-dimensional contact metric manifolds with ▽ξτ=0

Abstract: We give the classification of the 3-dimensional contact metric manifolds satisfying ▽ξτ=0, which they have: harmonic curvature, or η-parralel Ricci tensor or cyclic η-parallel Ricci tensor.

Gradient K\"ahler-Ricci Solitons and Periodic Orbits

Abstract: We study Hamiltonian dynamics of gradient Kaehler-Ricci solitons that arise as limits of dilations of singularities of the Ricci flow on compact Kaehler manifolds. Our main result is that the underlying spaces of such gradient solitons must be Stein manifolds. Moreover, on all most all energy surfaces of the potential function f of such a soliton, the Hamiltonian vector field V_f of f, with respect to the Kaehler form of the gradient soliton metric, admits a periodic orbit. The latter should be impotant in the study of singularities of the Ricci flow on compact Kaehler manifolds in light of the ``little loop lemma'' principle due to the second author.

The identities of the algebraic invariants of the four-dimensional Riemann tensor. II

Abstract: This paper extends the investigation of the invariants of the Riemann tensor to include the invariants that are of odd degree in the trace-free Ricci tensor. It is shown that these invariants can be expressed in terms of 15 such invariants that are irreducible. As a consequence, it is possible to write down a complete set of invariants of the Riemann tensor. Several syzygies for these invariants have been found and examples of these are given. These syzygies suggest there may be several new syzygies of invariants with even degree in the trace-free Ricci tensor. A large number of these have also been found and are discussed in the paper.

REAL HYPERSURFACES WITH ξ-PARALLEL RICCI TENSOR IN A COMPLEX SPACE FORM

Abstract: We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽S = 0 and Sξ = ξ for a smooth function , then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

Abstract: In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

Abstract: In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kahler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ℂn. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].

A Remark on the Geometry of Two-Dimensional Anisotropic Non-Linear Sigma-Models

Abstract: One discusses here the connection between \sigma-model gauge anomalies and the existence of a connection with torsion that does not flatten the Ricci tensor of the target manifold, by considering a number of non-symmetric coset spaces. The influence of an eventual anisotropy on a certain direction of the target manifold is also contemplated.

The einstein-kähler metrics with explicit forms on a class of egg domains

Abstract: The Einstein-Kahler metrics with -1 Ricci curvature for a class of egg domains in are given in the explicit forms. Under those metrics, it is very interesting that the holomorphic sectional curvatures are equal to the same constant -2/(n+1).