Showing papers on "Scalar curvature published in 1994"

Riemannian Geometry: A Modern Introduction

Abstract: 1. Riemannian manifolds 2. Riemannian curvature 3. Riemannian volume 4. Riemannian coverings 5. Surfaces 6. Isoperimetric inequalities (constant curvature) 7. The kinetic density 8. Isoperimetric inequalities (variable curvature) 9. Comparison and finiteness theorems.

Comparison Theorems in Riemannian Geometry

Abstract: The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method. The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work: The comparison theory for the Riccati ODE, which probably goes back to L.Green [15] and which was used more systematically by Gromov [20], the triangle inequality for the Riemannian distance, the method of support function by Greene and Wu [16],[17],[34], the maximum principle of E.Hopf, generalized by E.Calabi [23], [4], and the idea of critical points of the distance function which was first used by Grove and Shiohama [21]. We have tried to present the ideas completely without being too technical. These notes are based on a course which I gave at the University of Trento in March 1994. It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures. We also thank Evi Samiou and Robert Bock for many valuable corrections.

Embedding Riemannian manifolds by their heat kernel

Abstract: By embedding a class of closed Riemannian manifolds (satisfying some curvature assumptions and with diameter bounded from above) into the same Hilbert space, we interpret certain estimates on the heat kernel as giving a precompactness theorem on the class considered.

Extremal Kähler metrics and complex deformation theory

Abstract: Let (M, J, g) be a compact Kahler manifold of constant scalar curvature. Then the Kahler class [ω] has an open neighborhood inH 1,1 (M, ℝ) consisting of classes which are represented by Kahler forms of extremal Kahler metrics; a class in this neighborhood is represented by the Kahler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [ω] is “nondegenerate,” every small deformation of the complex manifold (M, J) also carries Kahler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kahler metrics on certain compact complex surfaces.

The Universality of vacuum Einstein equations with cosmological constant

Abstract: It is shown that for a wide class of analytic Lagrangians, which depend only on the scalar curvature of a metric and a connection, the application of the so called `Palatini formalism', i.e. treating the metric and the connection as independent variables, leads to `universal' equations. If the dimension n of spacetime is greater than two these universal equations are vacuum Einstein equations with cosmological constant for a generic Lagrangian and are suitably replaced by other universal equations at degenerate points. We show that degeneracy takes place in particular for conformally invariant Lagrangians and we prove that their solutions are conformally equivalent to solutions of Einstein's equations. For two-dimensional spacetimes we find instead that the universal equation is always the equation of constant scalar curvature; in this case the connection is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vector field ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their degenerate points.

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:

Fano Manifolds, Contact Structures, and Quaternionic Geometry

Abstract: Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kahler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kahler manifold (M4n, g). If Z also admits a second complex contact structure , then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

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.

Riemannian metrics with large

Abstract: We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian. Let (Mn, g) be a compact, connected Riemannian manifold of n dimensions. The Laplacian Ag acting on functions on M has discrete spectrum. Let A1 (g) denote the smallest positive eigenvalue of Ag . Hersch [5] proved that AIl(g)vol(S2, g) 3, the sphere Sn admits metrics of volume one with A1 arbitrarily large [3, 6]. Bleecker conjectured in [3] that such metrics exist on every manifold Mn if n > 3. In this note we give a very simple proof of Bleecker's conjecture using known examples and quite general principles. The same result has been proved independently by Xu [7] by a construction similar to ours. His argument, however, is much harder than our proof. Theorem 1. Every compact manifold Mn with n > 3 admits metrics g of volume one with arbitrarily large AI (g) . Proof. The idea of the proof is very simple. We take a metric go on Sn with vol(Sn, go) = 1 and AL (go) > k + 1, where k is a large constant. We excise from Sn a very small ball B(p, t) = B. and form the connected sum of Sn with M. The resulting manifold is diffeomorphic to M and has a submanifold Q, with smooth boundary, naturally identified with Sn \ Bi . Let g1 be an Received by the editors February 10, 1993. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 53C21. This work was done while the second author enjoyed the hospitality of Forschungsinstitut fur Mathematik at ETH Zurich. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

Quadratic divergence of geodesics in CAT(0) spaces

Abstract: A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.

Computation of self-similar solutions for mean curvature flow

Abstract: We describe a numerical algorithm to compute surfaces that are approximately self-similar under mean curvature flow. The method restricts computation to a two-dimensional subspace of the space of embedded manifolds that is likely to contain a self-similar solution. Using the algorithm, we recover the self-similar torus of Angenent and find several surfaces that appear to approximate previously unknown self-similar surfaces. Two of them may prove to be counterexamples to the conjecture of uniqueness of the weak solution for mean curvature flow for surfaces.

Almost Hermitian Geometry

Abstract: An algebraic study is made of the torsion and curvature of almost-Hermitian manifolds with emphasis on the space of curvature tensors orthogonal to those of Kahler metrics.

A density function and the structure of singularities of the mean curvature flow

Abstract: We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ℝ n+1, primarily in the boundaryless setting. We concentrate on the so-called “Type I” case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of “slow-forming singularities” of the flow. The main new ingredient introduced, to address these issues, is a certain “density function”, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.

Manifolds of Positive Scalar Curvature

Abstract: We survey the status of the problem of determining which differentiable manifolds (without boundary) have Riemannian metrics of positive scalar curvature. Of course, if the manifold is non-compact, one requires the metric to be complete.

Gravity from non-commutative geometry

Abstract: We introduce the linear connection in the non-commutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant, and apart from that we find no other dynamic fields in the model. Finally we discuss an example solution of flat linear connection, with the non-trivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the standard model, with the scaling factor corresponding to the Weinberg angle.

Addendum: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature

Abstract: is finite. On the other hand the argument at the beginning of Proposition 2.1 in [E] shows that if Al (B) is finite then Q(M, OM) is finite. Jin Zhiren pointed out to me that the Sobolev quotient Q(M, O9M) can be -oo. This is the case if we delete a small geodesic ball on a compact manifold without boundary with negative scalar curvature. More generally the Sobolev quotient is -oo if the first eigenvalue for the conformal Laplacian, with respect to Dirichlet boundary condition, is negative. In order to see that let p1 be the first eigenfunction for the problem

On the Volume of Unit Tangent Spheres in a Pinsler Manifold

Abstract: Motivated by some issues which enter into the Gauss-Bonnet-Chern theorem in Finsler geometry, this paper studies the unit tangent sphere (or indicatrix) Ix M at each point x of a Pinsler manifold M. We demonstrate that the volume of ImM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x. This contrasts sharply with the situation in Riemannian geometry. We also express the derivative of such volume function in terms of the second curvature tensor of the Chern connection. In particular, we find that this function is constant on Landsberg spaces (though that constant need not be equal to the value taken by Riemannian manifolds).

Yamabe metrics of positive scalar curvature and conformally flat manifolds

Abstract: Let CY(n,μ, R0 be the class of compact connected smooth manifolds M of dimension n ⩾ 3 and with Yamabe metrics g of unit volume such that each (M, g) is conformally flat and satisfies μ(M,[g]) ⩾μ 0 > 0, ∫ M |E g | n 2 dv g ⩽R 0 , where [g], μ(M,[g]) and Eg denote the conformal class of g, the Yamabe invariant of (M,[g]) and the traceless part of the Ricci tensor of g, respectively. In this paper, we study the boundary ACY(n, μ0, R0 of CY(n, μ0, R0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in ACY(n, μ0, R0) is a compact metric space (X,d). In particular, if (X,d) is not a point, then it has a structure of smooth manifold outside a finite subset S, and moreover, on F\ S there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d.

The Mixmaster Cosmological Metrics

Abstract: This paper begins with a short presentation of the Bianchi IX or “Mixmaster” cosmological model, and some ways of writing the Einstein equations for it. There is then an interlude describing how I came to a study of this model, and then a report of some mostly unpublished work from a Ph. D. thesis of D. M. (Prakash) Chitre relating approximate solutions to geodesic flows on finite volume negative curvature Riemannian manifolds, for which he could quote results on ergodicity. A final section restates studies of a zero measure set of solutions which in first approximation appear to have only a finite number of Kasner epochs before reaching the singularity. One finds no plausible case for such behavior in better approximations.

Spherical means on compact Riemannian manifolds of negative curvature

Abstract: We show that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds. This measure is related to the unique invariant measure for the strong unstable foliation, as well as the Patterson-Sullivan measure at infinity. It turns out to be invariant under the geodesic flow if and only if the mean curvature of the horospheres is constant. We use this measure in the study of rigidity problems.

Eingenvalues of the Dirac operator on compact Kähler manifolds

Abstract: Kahlerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kahler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kahlerian twistor operator. It is also proved that, on a Kahler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.