Showing papers on "Scalar curvature published in 1997"

Riemannian Manifolds: An Introduction to Curvature

Abstract: What Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.

Kähler-Einstein metrics with positive scalar curvature

Abstract: In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kahler manifolds, provided that the partial C 0-estimate posed in [T6] is true.

Ricci curvature and volume convergence

Abstract: The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and "almost extreme triangles" (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromov's conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

Nonpositive Curvature: Geometric and Analytic Aspects

Abstract: 1 Introduction.- 1.1 Examples of Riemannian manifolds of negative or nonpositive sectional curvature.- Appendix to 1.1: Symmetric spaces of noncompact type.- 1.2 Mordell and Shafarevitch type problems.- 1.3 Geometric superrigidity.- 2 Spaces of nonpositive curvature.- 2.1 Local properties of Riemannian manifolds of nonpositive sectional curvature.- 2.2 Nonpositive curvature in the sense of Busemann.- 2.3 Nonpositive curvature in the sense of Alexandrov.- 3 Convex functions and centers of mass.- 3.1 Minimizers of convex functions.- 3.2 Centers of mass.- 3.3 Convex hulls.- 4 Generalized harmonic maps.- 4.1 The definition of generalized harmonic maps.- 4.2 Minimizers of generalized energy functional.- 5 Bochner-Matsushima type identities for harmonic maps and rigidity theorems.- 5.1 The Bochner formula for harmonic one-forms and harmonic maps.- 5.2 A Matsushima type formula for harmonic maps.- 5.3 Geometrie superrigidity.

Motion by Mean Curvature from the Ginzburg-Landau Interface Model

Abstract: We consider the scalar field φ t with a reversible stochastic dynamics which is defined by the standard Dirichlet form relative to the Gibbs measure with formal energy . The potential V is even and strictly convex. We prove that under a suitable large scale limit the φ t -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the -field.

Estimates of the conformal scalar curvature equation via the method of moving planes

Abstract: In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal equation where K(x) is a positive continuous function, Z is a compact subset of , and g satisfies that is nonincreasing for t > 0. Assuming that the order of flatness at critical points of K on Z is no less than , we prove that, through the application of the method of moving planes, the inequality holds for any solution of (0.1) with Cap(Z) = 0. By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy Assume that the order of flatness at critical points of K is no less than n - 2; then the inequality holds for R ≤ 1. We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4). © 1997 John Wiley & Sons, Inc.

Kahler geometry of toric varieties and extremal metrics

Abstract: Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply and explicitly. Finally, a curious combinatorial formula for convex polytopes, that follows from the relation between the total integral of the scalar curvature and the wedge product of the first Chern class with a suitable power of the Kahler class, is presented.

Relative volume comparison with integral curvature bounds

Abstract: In this paper we shall generalize the Bishop-Gromov relative volume comparison estimate to a situation where one only has an integral bound for the part of the Ricci curvature which lies below a given number. This will yield several compactness and pinching theorems.

A priori estimates for prescribing scalar curvature equations

Abstract: We obtain a priori estimates for solutions to the prescribing scalar curvature equation (1) R(x)n-2 on Sn for n > 3. There have been a series of results in this respect. To obtain a priori estimates people required that the function R(x) be positive and bounded away from 0. This technical assumption has been used by many authors for quite a few years. It is due to the fact that the standard blowing-up analysis fails near R(x) = 0. The main objective of this paper is to remove this well-known assumption. Using the method of moving planes, we are able to control the growth of the solutions in the region where R is negative and in the region where R is small, and thus obtain a priori estimates on the solutions of (1) for a general function R which is allowed to change signs.

Are Galactic Rotation Curves Really Flat

Abstract: In this paper we identify an apparently previously unappreciated regularity in the systematics of galactic rotation curves; namely, we find that at the last detected points in galaxies of widely varying luminosity, the centripetal acceleration is found to have the completely universal form (v2/c2R)last = γ0/2 + γ*N*/2 + β*N*/R2, where γ0 and γ* are new universal constants, β* is the Schwarzschild radius of the Sun, and N* is the total amount of visible matter in each galaxy. This regularity points to a possible role for the linear potentials associated with conformal gravity, with the galaxy-independent γ0 term being found not to be generated from within individual galaxies at all but rather to be of cosmological origin, being due to the global Hubble flow of a necessarily spatially open universe of 3-space scalar curvature k = -(γ0/2)2 = -2.3 × 10-60 cm-2.

The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature

Abstract: In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.

Yamabe constants and the perturbed Seiberg–Witten equations

Abstract: Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-K\"ahler 4-manifolds.

The Riemannian Goldberg–Sachs Theorem

Abstract: The paper contains a description of compact Hermitian complex surfaces whose Riemannian Ricci tensor is of type (1,1). This in turn comes as a consequence of a Riemannian version of the well-known (generalized) Goldberg–Sachs theorem of the General Relativity. A complete proof of the Riemannian version is given in the framework of "classical" Hermitian geometry. The paper includes some more results also pertaining to "Riemannian Goldberg–Sachs theory", as well as a "dual theory" involving the Penrose operator.

Spacelike hypersurfaces of constant mean curvature and calabi-bernstein type problems

Abstract: Spacelike graphs of constant mean curvature over compact Riemannian manifolds in Lorentzian manifolds with constant sectional curvature are studied. The corresponding Calabi-Bernstein type problems are stated. In the case of nonpositive sectional curvature all their solutions are obtained, and for positive sectional curvature well-known results are extended.

Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space

Abstract: Spacelike hypersurfaces with prescribed mean curvature have played a major role in the study of Lorentzian manifolds Maximal mean curvature zero hypersurfaces were used in the rst proof of the positive mass theorem Constant mean curvature hypersurfaces provide convenient time gauges for the Einstein equations For a survey of results we refer to In and it was shown that entire solutions of the maximal surface equation

A Penrose-like Inequality for the Mass of Riemannian Asymptotically Flat Manifolds

Abstract: We prove an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature. Our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere (as in the original Penrose conjecture) and some nomalized Sobolev ratio. As expected, the equality case is achieved if and only if the metric is that of a standard spacelike slice in the Schwarzschild space.

Convergence Theorems in Riemannian Geometry

Abstract: This is a survey on the convergence theory developed rst by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, including Anderson's generalizations to the case where all one has is bounded Ricci curvature. The exposition is streamlined by the introduction of a norm for riemannian manifolds, which makes the theory more like that of Holder and Sobolev spaces.

Construction of Manifolds of Positive Ricci Curvature with Big Volume and Large Betti Numbers

Abstract: It is shown that a connected sum of an arbitrary number of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of Sha–Yang and Anderson, with volume bounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform euclidean volume growth and almost contain a line, thus showing topological instability of the splitting theorem of Cheeger–Gromoll, even in the presence of the lower volume bound. In the absence of such a bound, the topological instability was earlier shown by Anderson; metric stability holds, even without the volume bound, by the recent work of Colding–Cheeger.

The Topoligical Sphere Theorem for Complete Submanifolds

Abstract: A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold Mn in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mn has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k-currents, we eliminate Hk (Mn, Z) for all 1 ` k ` n − 1.We then observe that the fundamental group of Mn is trivial. It should be emphasized that our result is optimal.

Conformal vector fields on pseudo-Riemannian spaces

Abstract: We study conformal vector fields on pseudo-Riemannian manifolds, in particular on Einstein spaces and on spaces of constant scalar curvature. A global classification theorem for conformal vector fields is obtained which are locally gradient fields. This includes the case of a positive metric as well as the case of an indefinite metric.

Scalar curvature and the Thurston norm

Abstract: the supremum being taken over all connected, oriented surfaces Σ embedded in Y whose genus g is at least 2 [8]. If Y contains spheres or non-separating tori, the definition is extended by declaring that α has infinite norm if it has non-zero pairing with any sphere or torus. It has been noted by various authors that there is a connection between the genus of embedded surfaces and solutions of the SeibergWitten monopole equations. In the present context, the result can be phrased as follows. The monopole equations depend on the choice of a metric h on Y and a Spinc structure c, to which we can associate the class c1(c), the first Chern class of the associated spin bundle. The result then states that, if the dual Thurston norm of c1(c) is greater than 1, then there exists a metric h on Y for which the corresponding monopole equations admit no solution. A fact that lies rather deeper is that the connection between the Thurston norm and the monopole equations is sharp, in the following sense, at least in the case that Y is irreducible (that is, every sphere bounds a ball in Y ). Let us say that a cohomology class α ∈ H2(Y ;R) is a monopole class if there is a Spinc structure c, with c1(c) = α over the reals, such that the corresponding monopole equations have a solution for all Riemannian metrics h on Y . Then we have:

Volume-preserving mean curvature flow of rotationally symmetric surfaces

Abstract: A rotationally symmetric n-dimensional surface in \({\Bbb R}^{n+1}\), of enclosed volume V and with boundary in two parallel planes, is evolving under volume-preserving mean curvature flow. For large volume V, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to a constant mean curvature surface.

Foliations on Riemannian Manifolds and Submanifolds

Abstract: This text's key issue is the role of a Riemannian curvature in studies of manifolds and submanifolds with foliations. The results of many geometers are discussed, but this book principally focuses on the Riemannian geometry of manifolds and submanifolds with generators that have non-negative curvature, the main idea being that such manifolds decompose into a direct product when the dimension of leaves is sufficiently large.

Physical states of the quantum conformal factor

Abstract: The conformal factor of the spacetime metric becomes dynamical due to the trace anomaly of matter fields. Its dynamics is described by an effective action which we quantize by canonical methods on the Einstein universe R{times}S{sup 3}. We find an infinite tower of discrete states which satisfy the constraints of quantum diffeomorphism invariance. These physical states are in one-to-one correspondence with operators constructed by integrating integer powers of the Ricci scalar. {copyright} {ital 1997} {ital The American Physical Society}

A note on Fischer-Marsden’s conjecture

Abstract: In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere. Let M denote the set of smooth Riemannian metrics on an n-dimensional closed manifold M whose derivatives are L2-integrable. Then for any g E M, its scalar curvature Rg is an element in the space WV of C? functions. From the formula for Rg in local coordinates, we see that the scalar curvature map from M to WV defines a quasi-linear differential operator of second order. The derivative R' at g E M is given by (1) RI (h) =-A9 (tr9h) + 686g (h) -g(Ricg, h) where 6 is the divergence operator on the symmetric p-tensor on M, Ricg is the Ricci curvature tensor of g, A is the Laplacian, and 6* is the formal adjoint of 8. In the Riemannian case, if {ei}in=1 is a local orthonormal basis of vector fields, then