Showing papers on "Scalar curvature published in 1991"

The Ricci flow on the 2-sphere

Abstract: The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric Thus one can con-sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met-ric The primary interest in this question is not so much to give a newproof of the uniformization theorem, but rather to understand nonlinearparabolic equations better, especially those arising in differential geome-try A sufficiently deep understanding of parabolic equations should yieldimportant new results in Riemannian geometryThe question in the preceding paragraph has been studied by RichardHamilton [3] and Brad Osgood, Ralph Phillips and Peter Sarnak [6] In [3],Hamilton studied the following equation, which we refer to as

Hyperkähler and quaternionic Kähler geometry

Abstract: A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler A similar result is proved for 8-manifolds HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative) This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds The twistor space structure of the projectivised nilpotent orbits is studied

Constant mean curvature surfaces and integrable equations

Abstract: CONTENTS Introduction Chapter I. Compact surfaces of constant mean curvature 1. Differential equations of constant mean curvature surfaces 2. Sphere. Hopf theorem 3. Torus. Analytic formulation of the problem 4. Surfaces of higher genus Chapter II. Constant mean curvature tori 5. Doubly periodic solutions of the equation 6. The Baker-Akhiezer function. Analytic properties 7. The Baker-Akhiezer function. Explicit formulae 8. Reality 9. Formula for immersion 10. Periodicity conditions 11. Area 12. Wente tori 13. Singular spectral curves Chapter III. Constant mean curvature surfaces in S3 and H314. Equations of constant mean curvature surfaces in S3 and H3 15. Constant mean curvature spheres in S3 and H3 16. Constant mean curvature tori in S3 17. Minimal tori in S3 and Willmore tori 18. Constant mean curvature tori in H3 19. Minimal surfaces of higher genus in S3 Appendix. Painleve planes of constant mean curvature References

The riemannian manifold of all riemannian metrics

Abstract: In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like curvature” and ”scalar like curvature”. The pointwise character mentioned above allows us in section 3, to solve explicitly the geodesic equation and to obtain the domain of definition of the

Markov invariant geometry on manifolds of states

Abstract: This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers. Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states. A description is given of all invariant Riemannian metrics on manifolds of sectorial states. The equations of the geodesies for the entire family of invariant linear connections Δ=γΔ, γ∈ IR, are integrated on sets of classical probability distributions. A description is given of the protective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor γ(γ−1), with the Riemann-Christoffel curvature tensor.

On isotropy irreducible Riemannian manifolds

Abstract: A connected Riemannian manifold (M, g) is said to be isotropy irreducible if for each point p E M the isotropy group lip, i.e. all isometries of g fixing p, acts irreducibly on TpM via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By Schur's lemma the metric g is unique up to scaling among all metrics with the same isometry group. By the same argument, the Ricci tensor of g must be proportional to g, i.e. g is an Einstein metric. Furthermore, according to a theorem of Takahashi [Ta], every eigenspace of the Laplace operator of (M, g) with eigenvalue A:#0 and of dimension k+ 1 gives rise to an isometric minimal immersion into Sk(r) with r2=dimM/~, by using the eigenfunctions as coordinates (see Li [L] and w 6 of this paper for further properties of these minimal immersions). By a theorem of D. Bleecker [BI], these metrics can also be characterised as being the only metrics which are critical points for every natural functional on the space of metrics of volume 1 on a given manifold. From the definition it follows easily that the isometry group of g must act transitively on M. Hence (M, g) is also a Riemannian homogeneous space. Conversely, we can define a connected effective homogeneous space G/H to be isotropy irreducible if H is compact and Adn acts irreducibly on ~/~. Given an isotropy irreducible homogeneous space G/H, there exists a G-invariant metric g, unique up to scaling, such that (M, g) is isotropy irreducible in the first sense. But if we start with a Riemannian manifold (M, g) which is isotropy irreducible, it can give rise to several

Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Abstract: Assuming the existence of a real torus acting through holomorphic isometries on a Kahler manifold, we construct an ansatz for Kahler-Einstein metrics and an ansatz for Kahler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kahler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kahler metric. The superposition contains Kahler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP n. We also build such Kahler geometries on Kahler quotients of higher cohomogeneity.

Cubic recursive division with bounded curvature

Abstract: Previous cubic recursive division schemes have had either zero or unbounded curvature at the singular points. This paper describes a formulation which gives bounded non-zero curvature and close to curvature continuity. Although it was claimed at the conference that this formulation was C2, this is not (quite) true.

The Estimate of the First Eigenvalue of a Compact Riemannian Manifold

Abstract: The main theorem proved in this chapter is: Let M be a compact Riemannian manifold with nonnegative Ricci curvature. Then the first eigenvalue −λ1 of the Laplace operator of M satisfies λ1≥π2/ d2 , where d denotes the diameter of M. This estimate improves the recent results due to S. T. Yau and P. Li [1, 2] and gives the best estimate for this kind of manifold.

On Space-Like Hypersurfaces with Constant Mean Curvature of a Lorentz Space Form

Abstract: We study complete space-like hypersurfaces with constant mean curvature of a Lorentz space form.

Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set

Abstract: We consider complete manifolds with Ricci curvature nonnegative outside a compact set and prove that the number of ends of such a manifold is finite and in particular, we give an explicit upper bound for the number.