Showing papers on "Scalar curvature published in 2002"

The entropy formula for the Ricci flow and its geometric applications

Abstract: We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

Curvature Quintessence

Abstract: The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order theories of gravity We can define effective pressure and energy density directly connected to the Ricci scalar of curvature of a generic fourth order theory and then ask for the conditions to get an accelerated expansion Exact accelerated expanding solutions can be achieved for several fourth order theories so that we get an alternative scheme to the standard quintessence scalar field, minimally coupled to gravity, usually adopted We discuss also conformal transformations in order to see the links of quintessence between the Jordan and Einstein frames

Scalar Curvature and Stability of Toric Varieties

Abstract: We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety. This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of KahlerEinstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of

Differential Geometry: Curves - Surfaces - Manifolds

Abstract: * Notations and prerequisites from analysis* Curves in $\mathbb{R}^n$* The local theory of surfaces* The intrinsic geometry of surfaces* Riemannian manifolds* The curvature tensor* Spaces of constant curvature* Einstein spaces* Solutions to selected exercises* Bibliography* List of notation* Index

Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature

Abstract: In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem.

Proximal Point Algorithm On Riemannian Manifolds

Abstract: In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.

Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds

Abstract: . The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.

The nature of singularities in mean curvature flow of mean-convex sets

Abstract: Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the mean curvature of ∂K is everywhere nonnegative (with respect to the inward unit normal) and is not identically 0. More generally, it means that Ft(K) is contained in the interior of K for t > 0, where Ft(K) is the set obtained by letting K evolve for time t under the level set mean curvature flow. As K evolves, it traces out a closed set K of spacetime: K = {(x, t) ∈ R ×R : x ∈ Ft(K)}. Also, there is associated to K a Brakke flow M : t 7→Mt of rectifiable varifolds. We call the pair (M,K) a mean-convex flow. Let X = (x, t) be a point in spacetime with t > 0. Suppose (xi, ti) is a sequence of points converging to X and λi is a sequence of positive numbers tending to infinity. Translate the pair M and K in spacetime by (y, τ) 7→ (y − xi, τ − ti) and then dilate parabolically by (y, τ) 7→ (λiy, λi τ) to get new flows Mi and Ki. The sequence (Mi,Ki) is called a blow-up sequence at X . General compactness theorems guarantee that this sequence will have subsequential limits. A subsequential limit (M′,K′) is called a limit flow. Here M′ : t ∈ (−∞,∞) 7→M ′ t

On the singularities of spaces with bounded Ricci curvature

Abstract: Abstract. ((Without Abstract)).

Cohomogeneity one manifolds with positive Ricci curvature

Abstract: One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.

Covariance and Fisher information in quantum mechanics

Abstract: Variance and Fisher information are ingredients of the Cramer-Rao inequality. We regard Fisher information as a Riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and Fisher information. In this approach we show that there is a kind of dual one-to-one correspondence between the candidates of the two concepts. We emphasize that Fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold.

Ricci flow on Kähler-Einstein surfaces

Abstract: In this paper, we prove that if M is a Kahler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kahler-Ricci flow converges to a Kahler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kahler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kahler Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kahler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.

Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension

Abstract: Let f:Σ1↦Σ2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ1×Σ2 by the mean curvature flow. Under suitable conditions on the curvature of Σ1 and Σ2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschitz initial data.

Curvature and function theory on riemannian manifolds

Abstract: Function theory on Euclidean domains in relation to potential theory, partial differential equations, probability, and harmonic analysis has been the target of investigation for decades. There is a wealth of classical literature in the subject. Geometers began to study function theory with the primary reason to prove a uniformization type theorem in higher dimensions. It was first proposed by GreeneWu and Yau to study the existence of bounded harmonic functions on a complete manifold with negative curvature. While uniformization in dimension greater than 2 still remains an open problem, the subject of function theory on complete manifolds takes on life of its own. The seminal work of Yau [Y1] provided a fundamental technique in handling analysis on noncompact, complete manifolds. It also opens up many interesting problems which are essential for the understanding of analysis on complete manifolds. Since Yau’s paper in 1975, there are many developments in this subject. The aim of this article is to give a rough outline of the history of a specific point of view in this area, namely, the interplay between the geometry – primarily the curvature – and the function theory. Throughout this article, unless otherwise stated, we will assume that M is an n-dimensional, complete, noncompact, Riemannian manifold without boundary. In this case, we will simply say that M is a complete manifold. One of the goal of this survey is to demonstrate, by way of known theorems, the two major steps which are common in many geometric analysis programs. First, we will show how one can use assumptions on the curvature to conclude function theoretic properties of the manifold M. Secondly, we will showed that function theoretic properties can in turn be used to conclude geometrical and topological statements about the manifold. In many incidents, combining the two steps will result in a theorem which hypothesizes on the curvature and concludes on either the topological, geometrical, or complex structure of the manifold. The references will not be comprehensive due to the vast literature in the subject. It is merely an indication of the flavor of the field for the purpose of whetting one’s appetite. As examples of areas not being discussed in this note are harmonic analysis (function theory) on symmetric spaces, Lie groups, and discrete groups. The contributors to this subject are Furstenberg, Varopoulos, Coulhon, Saloff-Coste, and

Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature

Abstract: The purpose of this paper is to evolve non-smooth Riemannian metric tensors by the dual Ricci-Harmonic map flow. This flow is equivalent (up to a diffeomorphism) to the Ricci flow. One application will be the evolution of metrics which arise in the study of spaces whose curvature is bounded from above and below in the sense of Aleksandrov, and whose curvature operator (in dimension three Ricci curvature) is non-negative. We show that such metrics may always be deformed to a smooth metric having the same properties in a strong sense. §

Projectively flat Finsler metrics of constant flag curvature

Abstract: Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

Some properties of the Schouten tensor and applications to conformal geometry

Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the kth elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ + k . We prove that this eigenvalue condition for k > n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.

A momentum construction for circle-invariant Kähler metrics

Abstract: Examples of Kahler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kahler metrics of constant scalar curvature by ODE methods. One fruitful idea-the Calabi ansatz-is to begin with an Hermitian line bundle p: (L, h) → (M, g M ) over a Kahler manifold, and to search for Kahler forms ω = p*ωM + dd c f(t) in some disk subbundle, where t is the logarithm of the norm function and f is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kahler metric g arising from the Calabi ansatz. This suggests geometric hypotheses (which we call a-constancy) to impose upon the base metric g M and Hermitian structure h in order that the scalar curvature of g be specified by solving an ODE. We show that σ-constancy is necessary and sufficient for the Calabi ansatz to work in the following sense. Under the assumption of σ-constancy, the disk bundle admits a one-parameter family of complete Kahler metrics of constant scalar curvature that restrict to g M on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of σ-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kahler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kahler metric on C 2 .

Selfdual Einstein Metrics with Torus Symmetry

Abstract: It is well-known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on • 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kahler quotients of • Pm—1 by an (m — 2)-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kahler quotients of • P2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

Some remarks on Finsler manifolds with constant flag curvature

Abstract: This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in CPn. The third remark is that there is a description of the Finsler metrics of constant curvature on S2 in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on S2 to construct a global Finsler metric of constant positive curvature on S2. The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1 > 2. It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S1·GL(n,R)structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. 1991 Mathematics Subject Classification. 53B40, 53C60, 58A15.

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Abstract: Guided by the Hopf fibration, a family (indexed by a positive constant ) of right invariant Riemannian metrics on the Lie group is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each . Each such Riemannian metric couples with the corresponding Killing field to produce a -global and explicit Randers metric on . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

Angle theorems for the Lagrangian mean curvature flow

Abstract: We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle $\alpha$ for the corresponding Lagrangian submanifold in the cross product space $L\times M$ satisfies $\text{osc}(\alpha)\le \pi$ . If one considers a 4-dimensional Kahler-Einstein manifold $\overline{M}$ of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that $L\subset\overline{M}$ is a compact oriented Lagrangian submanifold w.r.t. J such that the Kahler form $\overline{\kappa}$ w.r.t.K restricted to L is positive and $\text{osc}(\alpha)\le \pi$ , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. $\overline{\kappa}$ .

Sharp local isoperimetric inequalities involving the scalar curvature

Abstract: We provide sharp local isoperimetric inequalities on Riemannian manifolds involving the scalar curvature, and thus answer a question asked by Johnson and Morgan.

On the Yamabe problem and the scalar curvature problems under boundary conditions

Abstract: In this paper we prove some existence results concerning a problem arising in conformal differential geometry. Consider a smooth metric g onB = {x ∈ R : |x| < 1}, the unit ball onR, n ≥ 3, and let∆g, Rg, νg, hg denote, respectively, the Laplace-Beltrami operator, the scalar curvature of (B, g), the outward unit normal to∂B = Sn−1 with respect tog and the mean curvature of (Sn−1, g). Given two smooth functions R′ andh′, we will be concerned with the existence of positive solutionsu ∈ H 1(B) of   −4(n− 1) (n− 2)∆gu+ Rgu = R ′u n+2 n−2 , in B;

dS/CFT and spacetime topology

Abstract: Motivated by recent proposals for a de Sitter version of the AdS/CFT correspondence, we give some topological restrictions on spacetimes of de Sitter type, i.e., spacetimes with $\Lambda>0$, which admit a regular past and/or future conformal boundary. For example we show that if $M^{n+1}$, $n \ge 2$, is a globally hyperbolic spacetime obeying suitable energy conditions, which is of de Sitter type, with a conformal boundary to both the past and future, then if one of these boundaries is compact, it must have finite fundamental group and its conformal class must contain a metric of positive scalar curvature. Our results are closely related to theorems of Witten and Yau hep-th/9910245 pertaining to the Euclidean formulation of the AdS/CFT correspondence.

Minimal Hypersurfaces with Finite Index

Abstract: In an article of Cao-Shen-Zhu [C-S-Z], they proved that a complete, immersed, stable minimal hypersurface M of R with n ≥ 3 must have only one end. When n = 2, it was proved independently by do Carmo-Peng [dC-P] and FischerColbrie-Schoen [FC-S] that a complete, immersed, oriented stable minimal surface in R must be a plane. Later Gulliver [G] and Fischer-Colbrie [FC] proved that if a complete, immersed, minimal surface in R has finite index, then it must be conformally equivalent to a compact Riemann surface with finitely many punctures. Fischer-Colbrie actually proved this for minimal surfaces in a complete manifold with non-negative scalar curvature. In any event, a corollary is that if a complete, immersed, oriented minimal surface in R has finite index then it must have finitely many ends. The purpose of this paper is to generalize this result for finitely many ends to higher dimensional minimal hypersurfaces in Euclidean space (see Theorem 5). In fact, we will also show that the first L-Betti number of such a manifold must be finite. The strategy of Cao-Shen-Zhu was to utilize a result a Schoen-Yau [S-Y] asserting that a complete, stable minimal hypersurface of R cannot admit a non-constant harmonic function with finite Dirichlet integral. Assuming that M has more than one end, Cao-Shen-Zhu constructed a non-constant harmonic function with finite Dirichlet integral. This approach very much fits into the scheme studied by the first author and Tam in [L-T]. In fact, the authors showed that the number of non-parabolic ends of any complete Riemannian manifold is bounded above by the dimension of the space of bounded harmonic functions with finite Dirichlet integral. The proof of Cao-Shen-Zhu can be modified to show that each end of a complete, immersed, minimal submanifold must be non-parabolic. Due to this connection

Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds

Abstract: Let (M m , g) be a compact Riemannian manifold isometrically immersed in a simply connected space form (euclidean space, sphere or hyperbolic space). The purpose of this paper is to give optimal upper bounds for the first nonzero eigenvalue of the Laplacian of (M m , g) in terms of r-th mean curvatures and scalar curvature. As consequences, we obtain some rigidity results. In particular, we prove that if (M n ,g) is a compact hypersurface of positive scalar curvature immersed in R n+1 and if g is a Yamabe metric, then (M n ,g) is a standard sphere.

An Extension of The Classical Ribaucour Transformation

Abstract: We extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Bianchi gave a positive answer to the question of whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. Our main achievement is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature. 2000 Mathematical Subject Classification: 53B25, 58J72.