Showing papers on "Scalar curvature published in 2000"

An Introduction to Riemann-Finsler Geometry

Abstract: One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 1.0 Physical Motivations.- 1.1 Finsler Structures: Definitions and Conventions.- 1.2 Two Basic Properties of Minkowski Norms.- 1.2 A. Euler's Theorem.- 1.2 B. A Fundamental Inequality.- 1.2 C. Interpretations of the Fundamental Inequality.- 1.3 Explicit Examples of Finsler Manifolds.- 1.3 A. Minkowski and Locally Minkowski Spaces.- 1.3 B. Riemannian Manifolds.- 1.3 C. Randers Spaces.- 1.3 D. Berwald Spaces.- 1.3 E. Finsler Spaces of Constant Flag Curvature.- 1.4 The Fundamental Tensor and the Cartan Tensor.- * References for Chapter 1.- 2 The Chern Connection.- 2.0 Prologue.- 2.1 The Vector Bundle ?*TM and Related Objects.- 2.2 Coordinate Bases Versus Special Orthonormal Bases.- 2.3 The Nonlinear Connection on the Manifold TM \0.- 2.4 The Chern Connection on ?*TM.- 2.5 Index Gymnastics.- 2.5 A. The Slash (...)s and the Semicolon (...) s.- 2.5 B. Covariant Derivatives of the Fundamental Tensor g.- 2.5 C. Covariant Derivatives of the Distinguished ?.- * References for Chapter 2.- 3 Curvature and Schur's Lemma.- 3.1 Conventions and the hh-, hv-, vv-curvatures.- 3.2 First Bianchi Identities from Torsion Freeness.- 3.3 Formulas for R and P in Natural Coordinates.- 3.4 First Bianchi Identities from "Almost" g-compatibility.- 3.4 A. Consequences from the $$ dx^k \wedge dx^l $$ Terms.- 3.4 B. Consequences from the $$ dx^k \wedge \frac{1} {F}\delta y^l $$ Terms.- 3.4 C. Consequences from the $$ \frac{1} {F}\delta y^k \wedge \frac{1} {F}\delta y^l $$ Terms.- 3.5 Second Bianchi Identities.- 3.6 Interchange Formulas or Ricci Identities.- 3.7 Lie Brackets among the $$ \frac{\delta } {{\delta x}} $$ and the $$ F\frac{\partial } {{\partial y}} $$.- 3.8 Derivatives of the Geodesic Spray Coefficients Gi.- 3.9 The Flag Curvature.- 3.9 A. Its Definition and Its Predecessor.- 3.9 B. An Interesting Family of Examples of Numata Type.- 3.10 Schur's Lemma.- *References for Chapter 3.- 4 Finsler Surfaces and a Generalized Gauss-Bonnet Theorem.- 4.0 Prologue.- 4.1 Minkowski Planes and a Useful Basis.- 4.1 A. Rund's Differential Equation and Its Consequence.- 4.1 B. A Criterion for Checking Strong Convexity.- 4.2 The Equivalence Problem for Minkowski Planes.- 4.3 The Berwald Frame and Our Geometrical Setup on SM.- 4.4 The Chern Connection and the Invariants I, J, K.- 4.5 The Riemannian Arc Length of the Indicatrix.- 4.6 A Gauss-Bonnet Theorem for Landsberg Surfaces.- *References for Chapter 4.- Two Calculus of Variations and Comparison Theorems.- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature.- 5.1 The First Variation of Arc Length.- 5.2 The Second Variation of Arc Length.- 5.3 Geodesics and the Exponential Map.- 5.4 Jacobi Fields.- 5.5 How the Flag Curvature's Sign Influences Geodesic Rays.- *References for Chapter 5.- 6 The Gauss Lemma and the Hopf-Rinow Theorem.- 6.1 The Gauss Lemma.- 6.1 A. The Gauss Lemma Proper.- 6.1 B. An Alternative Form of the Lemma.- 6.1 C. Is the Exponential Map Ever a Local Isometry?.- 6.2 Finsler Manifolds and Metric Spaces.- 6.2 A. A Useful Technical Lemma.- 6.2 B. Forward Metric Balls and Metric Spheres.- 6.2 C. The Manifold Topology Versus the Metric Topology.- 6.2 D. Forward Cauchy Sequences, Forward Completeness.- 6.3 Short Geodesics Are Minimizing.- 6.4 The Smoothness of Distance Functions.- 6.4 A. On Minkowski Spaces.- 6.4 B. On Finsler Manifolds.- 6.5 Long Minimizing Geodesies.- 6.6 The Hopf-Rinow Theorem.- *References for Chapter 6.- 7 The Index Form and the Bonnet-Myers Theorem.- 7.1 Conjugate Points.- 7.2 The Index Form.- 7.3 What Happens in the Absence of Conjugate Points?.- 7.3 A. Geodesies Are Shortest Among "Nearby" Curves.- 7.3 B. A Basic Index Lemma.- 7.4 What Happens If Conjugate Points Are Present?.- 7.5 The Cut Point Versus the First Conjugate Point.- 7.6 Ricci Curvatures.- 7.6 A. The Ricci Scalar Ric and the Ricci Tensor Ricij.- 7.6 B. The Interplay between Ric and RiCij.- 7.7 The Bonnet-Myers Theorem.- *References for Chapter 7.- 8 The Cut and Conjugate Loci, and Synge's Theorem.- 8.1 Definitions.- 8.2 The Cut Point and the First Conjugate Point.- 8.3 Some Consequences of the Inverse Function Theorem.- 8.4 The Manner in Which cy and iy Depend on y.- 8.5 Generic Properties of the Cut Locus Cutx.- 8.6 Additional Properties of Cutx When M Is Compact.- 8.7 Shortest Geodesics within Homotopy Classes.- 8.8 Synge's Theorem.- *References for Chapter 8.- 9 The Cartan-Hadamard Theorem and Rauch's First Theorem.- 9.1 Estimating the Growth of Jacobi Fields.- 9.2 When Do Local Diffeomorphisms Become Covering Maps?.- 9.3 Some Consequences of the Covering Homotopy Theorem.- 9.4 The Cartan-Hadamard Theorem.- 9.5 Prelude to Rauch's Theorem.- 9.5 A. Transplanting Vector Fields.- 9.5 B. A Second Basic Property of the Index Form.- 9.5 C. Flag Curvature Versus Conjugate Points.- 9.6 Rauch's First Comparison Theorem.- 9.7 Jacobi Fields on Space Forms.- 9.8 Applications of Rauch's Theorem.- *References for Chapter 9.- Three Special Finsler Spaces over the Reals.- 10 Berwald Spaces and Szabo's Theorem for Berwald Surfaces.- 10.0 Prologue.- 10.1 Berwald Spaces.- 10.2 Various Characterizations of Berwald Spaces.- 10.3 Examples of Berwald Spaces.- 10.4 A Fact about Flat Linear Connections.- 10.5 Characterizing Locally Minkowski Spaces by Curvature.- 10.6 Szabo's Rigidity Theorem for Berwald Surfaces.- 10.6 A. The Theorem and Its Proof.- 10.6 B. Distinguishing between y-local and y-global.- *References for Chapter 10.- 11 Randers Spaces and an Elegant Theorem.- 11.0 The Importance of Randers Spaces.- 11.1 Randers Spaces, Positivity, and Strong Convexity.- 11.2 A Matrix Result and Its Consequences.- 11.3 The Geodesic Spray Coefficients of a Randers Metric.- 11.4 The Nonlinear Connection for Randers Spaces.- 11.5 A Useful and Elegant Theorem.- 11.6 The Construction of y-global Berwald Spaces.- 11.6 A. The Algorithm.- 11.6 B. An Explicit Example in Three Dimensions.- *References for Chapter 11 309.- 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem.- 12.0 Prologue.- 12.1 Characterizations of Constant Flag Curvature.- 12.2 Useful Interpretations of ? and E.- 12.3 Growth Rates of Solutions of E + ? E = 0.- 12.4 Akbar-Zadeh's Rigidity Theorem.- 12.5 Formulas for Machine Computations of K.- 12.5 A. The Geodesic Spray Coefficients.- 12.5 B. The Predecessor of the Flag Curvature.- 12.5 C. Maple Codes for the Gaussian Curvature.- 12.6 A Poincare Disc That Is Only Forward Complete.- 12.6 A. The Example and Its Yasuda-Shimada Pedigree.- 12.6 B. The Finsler Function and Its Gaussian Curvature.- 12.6 C. Geodesics Forward and Backward Metric Discs.- 12.6 D. Consistency with Akbar-Zadeh's Rigidity Theorem.- 12.7 Non-Riemannian Projectively Flat S2 with K = 1.- 12.7 A. Bryant's 2-parameter Family of Finsler Structures.- 12.7 B. A Specific Finsler Metric from That Family.- *References for Chapter 12 350.- 13 Riemannian Manifolds and Two of Hopf's Theorems.- 13.1 The Levi-Civita (Christoffel) Connection.- 13.2 Curvature.- 13.2 A. Symmetries, Bianchi Identities, the Ricci Identity.- 13.2 B. Sectional Curvature.- 13.2 C. Ricci Curvature and Einstein Metrics.- 13.3Warped Products and Riemannian Space Forms.- 13.3 A. One Special Class of Warped Products.- 13.3 B. Spheres and Spaces of Constant Curvature.- 13.3 C. Standard Models of Riemannian Space Forms.- 13.4 Hopf's Classification of Riemannian Space Forms.- 13.5 The Divergence Lemma and Hopf's Theorem.- 13.6 The Weitzenbock Formula and the Bochner Technique.- *References for Chapter 13.- 14 Minkowski Spaces, the Theorems of Deicke and Brickell.- 14.1 Generalities and Examples.- 14.2 The Riemannian Curvature of Each Minkowski Space.- 14.3 The Riemannian Laplacian in Spherical Coordinates.- 14.4 Deicke's Theorem.- 14.5 The Extrinsic Curvature of the Level Spheres of F.- 14.6 The Gauss Equations.- 14.7 The Blaschke-Santalo Inequality.- 14.8 The Legendre Transformation.- 14.9 A Mixed-Volume Inequality, and Brickell's Theorem.- * References for Chapter 14.

On the structure of spaces with Ricci curvature bounded below. I

Abstract: In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of

On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies

Abstract: For general relativistic spacetimes filled with irrotational ‘dust’ a generalized form of Friedmann's equations for an ‘effective’ expansion factor aD of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the ‘backreaction effect’ of inhomogeneities on the average expansion of the model. A universal relation between ‘backreaction’ and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to aD-2, the expansion law governing a generic domain can be found. However, as the general equations show, ‘backreaction’ acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

Abstract: On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝn (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity.

Curvature and symmetry of Milnor spheres

Abstract: Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

Canonical Metrics in Kahler Geometry

Abstract: 1 Introduction to Kahler manifolds.- 1.1 Kahler metrics.- 1.2 Curvature of Kahler metrics.- 2 Extremal Kahler metrics.- 2.1 The space of Kahler metrics.- 2.2 A brief review of Chern classes.- 2.3 Uniformization of Kahler-Einstein manifolds.- 3 Calabi-Futaki invariants.- 3.1 Definition of Calabi-Futaki invariants.- 3.2 Localization formula for Calabi-Futaki invariants.- 4 Scalar curvature as a moment map.- 5 Kahler-Einstein metrics with non-positive scalar curvature.- 5.1 The Calabi-Yau Theorem.- 5.2 Kahler-Einstein metrics for manifolds with c1(M) < 0.- 6 Kahler-Einstein metrics with positive scalar curvature.- 6.1 A variational approach.- 6.2 Existence of Kahler-Einstein metrics.- 6.3 Examples.- 7 Applications and generalizations.- 7.1 A manifold without Kahler-Einstein metric.- 7.2 K-energy and metrics of constant scalar curvature.- 7.3 Relation to stability.

Brane cosmologies without orbifolds

Abstract: We study the dynamics of branes in configurations where (1) the brane is the edge of a single anti--de Sitter (AdS) space and (2) the brane is the surface of a vacuum bubble expanding into a Schwarzschild or AdS-Schwarzschild bulk. In both cases we find solutions that resemble the standard Robertson-Walker cosmologies, although, in the latter, the evolution can be controlled by a mass parameter in the bulk metric. We also include a term in the brane action for the scalar curvature. This term adds a contribution to the low-energy theory of gravity which does not need to affect the cosmology, but which is necessary for the surface of the vacuum bubble to recover four-dimensional gravity.

Some new obstructions to minimal and Lagrangian isometric immersions Dedicated to Professor Tadashi Nagano on the occasion of his seventieth birthday

Abstract: 1. IntroductionThe main purpose of this paper is to introduce a new type of Riemannian curvature invariants and to show that these new invariants have interesting applications to several areas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions.One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean space (or more generally, in a space form). According to a well-known theorem of J. F. Nash, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimension.In order to study this fundamental problem, in view of Nash's theorem, it is natural to impose a suitable condition on the immersions. For instance, if one imposes the minimality condition on the immersions, it leads toPROBLEM 1. Given a Riemannian manifold M, what are the necessary conditions for M to admit a minimal isometric immersion in a Euclidean m-space Em?It is well-known that for a minimal submanifold in Em, the Ricci tensor satisfies Ric_??_0. For many years this was the only known general necessary Riemannian condition for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space.The main results of this article were presented at the 3rd Pacific Rim Geometry Conference held at Seoul, Korea in December 1996; also presented at the 922nd AMS meeting held at Detroit, Michigan in May 1997.

Motion of hypersurfaces by gauss curvature

Abstract: We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-smooth initial hypersurfaces, and develop upper and lower bounds on the speed and the curvature independent of initial conditions. Applications are given to the flow by affine normal and to the existence of non-spherical homothetically contracting solutions.

Higher-Order Corrections to the Effective Gravitational Action from Noether Symmetry Approach

Abstract: Higher-order corrections of the Einstein–Hilbert action of general relativity can be recovered by imposing the existence of a Noether symmetry to a class of theories of gravity where the Ricci scalar R and its d'Alembertian □ R are present. In several cases, it is possible to get exact cosmological solutions or, at least, to simplify the dynamics by recovering constants of motion. The main result is that a Noether vector seems to rule the presence of higher-order corrections of gravity.

Riemannian Geometry During the Second Half of the Twentieth Century

Abstract: Additional bibliography Riemannian geometry up to 1950 Comments on the main topics I, II, III, IV, V under consideration Curvature and topology The geometrical hierarchy of Riemann manifolds: Space forms The set of Riemannian structures on a given compact manifold: Is there a best metric? The spectrum, the eigenfunctions Periodic geodesics, the geodesic flow Some other Riemannian geometric topics of interest Bibliography Subject and notation index Name index

The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature

Abstract: Let (M,g) be an n dimensional compact, smooth, Riemannian manifold without boundary. For n = 2, the Uniformization Theorem of Poincare says that there exist metrics on M which are pointwise conformal to g and have constant Gauss curvature. For n > 3, the well known Yamabe conjecture states that there exist metrics on M which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture has been proved through the work of Yamabe [Y], Trudinger [T], Aubin [A], and Schoen [SI]. See Lee and Parker [LP] for a survey. See also Bahri and Brezis [BB], Bahri [B], and Schoen [S2-3] for works on the problem and related ones. Analogues of the Yamabe problem for compact Riemannian manifolds with boundary have been studied by Cherrier, Escobar, and others. In particular, Escobar proved in [E2] that a large class of compact Riemannian manifolds with boundary are conformally equivalent to one with constant scalar curvature and zero mean curvature on the boundary. See also [E3][E5] for related results. From now on in the paper, (M, g) denotes some smooth compact n dimensional Riemannian manifold with boundary, unless we specify otherwise. We use M to denote the interior of M, and dM the boundary of M. We use n — 2 d n — 2 La to denote An—c(n)Ra, where c(n) is — —, BQ to denote ——I—-—hQ, y y * 4(n -1) y du 2 y where u is the outward unit normal on dM with respect to 5, and hg to denote the mean curvature of dM with respect to the inner normal (balls in R have positive mean curvatures).

Collapsing Three-Manifolds Under a Lower Curvature Bound

Abstract: The purpose of this paper is to completely characterize the topology of three-dimensional Riemannian manifolds with a uniform lower bound of sectional curvature which converges to a metric space of lower dimension.

Analysis and geometry on manifolds with integral Ricci curvature bounds. II

Abstract: We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding’s volume convergence results and extend the Cheeger-Colding splitting theorem.

A Metric Characterization of Riemannian Submersions

Abstract: A map between metric spaces is called a submetry if it mapsballs of radius R around a point onto balls of the sameradius around the image point. We show that when the domain and targetspaces are complete Riemannian manifolds, submetries correspond toC 1,1 Riemannian submersions. We also study someconsequences of this fact, and introduce the notion of submetries with asoul.

Ricci Curvature, Minimal Volumes, and Seiberg-Witten Theory

Abstract: We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum of the L2-norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4-manifold minimizes volume among all metrics satisfying a point-wise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new non-existence results for Einstein metrics.

Compactification of a class of conformally flat 4-manifold

Abstract: In this paper we generalize Huber’s result on complete surfaces of finite total curvature. For complete locally conformally flat 4-manifolds of positive scalar curvature with Q curvature integrable, where Q is a variant of the Chern-Gauss-Bonnet integrand; we first derive the Cohn-Vossen inequality. We then establish finiteness of the topology. This allows us to provide conformal compactification of such manifolds.

Complete Riemannian manifolds with pointwise pinched curvature

Abstract: An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M n . We prove that if n≥4 and the curvature operator of M n is pointwise pinched, or if n=3 and the Ricci curvature of M 3 is pointwise pinched, then M n is compact.

Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature

Abstract: The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.

Local boundary rigidity of a compact Riemannian manifold with curvature bounded above

Abstract: This paper considers the boundary rigidity problem for a compact convex Riemannian manifold (M, g) with boundary O9M whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics g' on M there is a C3,,_ neighborhood of g such that g is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -as measured in M). More precisely, given any metric g' in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on O9M such that g' = *g. There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function. 1. STATEMENT OF THE RESULT The general boundary rigidity problem reads: to which extent is a Riemannian metric on a compact manifold with boundary determined from the distances between boundary points? More precisely, it can be formulated as follows. Let (M,g) be a compact Riemannian manifold with boundary OM. Let g' be another Riemannian metric on M. We say that g and g' have the same boundary distance-function if d1 (x, y) = d9, (x, y) for arbitrary boundary points x, y E OM, where d1 (resp. d9,) represents distance in M with respect to g (resp. g'). It is easy to give examples of pairs of metrics with the same boundary distance-function. Indeed, if : M -* M is an arbitrary diffeomorphism of M onto itself which is the identity on the boundary, then the metrics g and g' = *g have the same boundary distance-function. Here g' = ,og is the pull-back of g under ( (i.e., for arbitrary vectors ,1 E TxM we have ( -,r1' -( *71),(x)I where P: TxM * T (x) M is the differential of ( at x and (, ) (resp. (, )') is the inner product with respect to the metric g (resp. g')). We say that a compact Riemannian manifold is boundary rigid if this is the only type of nonuniqueness. More precisely, (M, g) is boundary rigid if, for every Received by the editors August 10, 1998. 2000 Mathematics Subject Classification. Primary 53C65, 53C22, 53C20; Secondary 58C25, 58C40. The first and third authors were supported by CRDF, Grant RM2-143. The first author was also supported by NSF grants #MDS-9505175 and #MDS-9626232. The third author was also supported by INTAS RFBR, Grant 95-0763. ?)2000 American Matlhematical Society 3937 This content downloaded from 207.46.13.167 on Thu, 21 Jul 2016 04:20:00 UTC All use subject to http://about.jstor.org/terms 3938 C. B. CROKE, N. S. DAIRBEKOV. AND V. A. SHARAFNJTDINOV Riemannian metric g' on M1with the same boundary distance-function as g, there is a diffeomorphism o: M ---> M which is the identity on the boundary and for which g' 0} of in-ward and outward vectors. Here v is the unit outward norn al to OM1. Given (x, () C QMVI, we denote by K(x, () the maximum of the sectional curvatures of all two-planes oC T1 M such that c a. For (M, g) a CDRM, we define the following invariant:

On fundamental groups of manifolds of nonnegative curvature

Abstract: We will characterize the fundamental groups of compact manifolds of (almost) nonnegative Ricci curvature and also the fundamental groups of manifolds that admit bounded curvature collapses to manifolds of nonnegative sectional curvature. Actually it turns out that the known necessary conditions on these groups are sufficient as well. Furthermore, we reduce the Milnor problem—are the fundamental groups of open manifolds of nonnegative Ricci curvature finitely generated?—to manifolds with abelian fundamental groups. Moreover, we prove for each positive integer n that there are only finitely many non-cyclic, finite, simple groups acting effectively on some complete n -manifold of nonnegative Ricci curvature. Finally, sharping a result of Cheeger and Gromoll [6], we show for a compact Riemannian manifold (M,g 0 ) of nonnegative Ricci curvature that there is a continuous family of metrics (g λ ),λ∈[0,1] such that the universal covering spaces of (M,g λ ) are mutually isometric and (M,g 1 ) is finitely covered by a Riemannian product N×T d , where T d is a torus and N is simply connected.

Kato constants in Riemannian geometry

Abstract: We derive improvements of Kato’s inequality for sections of an irreducible SO(n)or Spin(n)-bundle V in the category of Riemannian manifolds, under the assumption that the section solves a first-order equivariant injectively elliptic system D. An explicit, general (covering all V and D) formula is given for the improved Kato constant.

Noncollapsing examples with positive Ricci curvature and infinite topological type

Abstract: Abstract. ((Without Abstract)).

On the Structure of Solutions to the Static Vacuum Einstein Equations

Abstract: (0.2) These equations are the simplest equations for Ricci-flat 4-manifolds. They have been extensively studied in the physics literature on classical relativity,where the solutions represent space-times outside regions of matter which are translation and reflection invariant in the time direction t. However,with the exception of some notable instances,(c.f. Theorem 0.1 below),many of the global properties of solutions have not been rigorously examined,either from mathematical or physical points of view,c.f. [Br] for example. This paper is also motivated by the fact that solutions of the static vacuum equations arise in the study of degenerations of Yamabe metrics (or metrics of constant scalar curvature) on 3-manifolds,c.f. [A1]. Because of this and other related applications of these equations to the geometry of 3-manifolds,we are interested in general mathematical aspects of the equations and their solutions which might not be physically relevant; for example,we allow solutions with negative mass. In this paper,we will be mostly concerned with the geometry of the 3manifold solutions (M, g, u) of (0.1),(i.e. the space-like hypersurfaces),and not with the 4-manifold metric. Thus,the choice of Riemannian or Lorentzian geometry on N in (0.2) will play no role. This considerably simplifies the discussion of singularities and boundary structure,but still allows for a large variety of behaviors; c.f. [ES] for a survey on singularities of space-times. Obviously,there are no non-flat solutions to (0.1) on closed manifolds,and so it will be assumed that M is an open,connected oriented 3-manifold. Let ¯ M

Hypersurfaces of prescribed curvature in Lorentzian manifolds

Abstract: The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved, provided there are barriers. . 0 INTRODUCTION Consider the problem of finding a closed hypersurface of prescribed curvature F in a complete (n+1)-dimensional manifold N. To be more precise, let Ω be a connected open subset of N, f ∈ C2,α(Ω̄), F a smooth, symmetric function defined in an open cone Γ ⊂ Rn, then we look for a hypersurface M ⊂ Ω such that F|M = f(x) for all x ∈ M, (0.1) where F|M means that F is evaluated at the vector (κi(x)) the components of which are the principal curvatures of M. The prescribed function f should satisfy natural structural conditions, e.g., if Γ is the positive cone and the hypersurface M is supposed to be convex, then f should be positive, but no further, merely technical, conditions should be imposed. If N is a Riemannian manifold, then the problem has been solved in the case when F = H, the mean curvature, where in addition n had to be small, and N conformally flat, cf. [7], and for curvature functions F of class (K), no restrictions on n, cf. [4, 6]. We also refer to [5], where more special situations are considered, and the bibliography therein. 1125 Indiana University Mathematics Journal c ©, Vol. 49, No. 3 (2000)

Three-Dimensional Lorentz Metrics and Curvature Homogeneity of Order One

Abstract: In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k ∈ N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.

Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups

Abstract: In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.