Showing papers on "Scalar curvature published in 1999"

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Abstract: We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds etc. which are related to recurrence and non-explosion.

On average properties of inhomogeneous fluids in general relativity I: dust cosmologies

Abstract: For general relativistic spacetimes filled with irrotational `dust' a generalized form of Friedmann's equations for an `effective' expansion factor $a_D (t)$ 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 $a_D^{-2}$, 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.

Classification of solutions of higher order conformally invariant equations

Abstract: Recently, there have been much analytic work on the conformally invariant operators as well as its associated differential equations. A well known second order conformally invariant operator comes from the Yamabe problem or, more generally, the problem of prescribed scalar curvature. Given a smooth positive function K defined on a compact Riemannian manifold (M, g0) of dimension n ≥ 2, we ask whether there exists a metric g conformal to g0 such that K is the scalar curvature of the new metric g. Let g = eg0 for n = 2 or g = u 4 n−2 g0 for n ≥ 3, then the problem is reduced to find solutions of the following nonlinear elliptic equations:

Gauss curvature flow: the fate of the rolling stones

Abstract: We prove Firey’s 1974 conjecture that convex surfaces moving by their Gauss curvature become spherical as they contract to points.

Connectedness of the boundary in the AdS / CFT correspondence

Abstract: Let M be a complete Einstein manifold of negative curvature, and assume that (as in the AdS/CFT correspondence) it has a Penrose compactification with a conformal boundary N of positive scalar curvature. We show that under these conditions, H_n(M;Z) = 0 and in particular N must be connected. These results resolve some puzzles concerning the AdS/CFT correspondence.

Mean curvature flow singularities for mean convex surfaces

Abstract: We study the evolution by mean curvature of a smooth n–dimensional surface ${\cal M}\subset{\Bbb R}^{n+1}$ , compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case $n=2$ .

Refined asymptotics for constant scalar curvature metrics with isolated singularities

Abstract: We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \(\), in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

Yamabe type equations on three dimensional riemannian manifolds

Abstract: A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

An Introduction to the Analysis of Paths on a Riemannian Manifold

Abstract: Brownian motion in Euclidean space Diffusions in Euclidean space Some addenda, extensions, and refinements Doing it on a manifold, an extrinsic approach More about extrinsic Riemannian geometry Bochner's identity Some intrinsic Riemannian geometry The bundle of orthonormal frames Local analysis of Brownian motion Perturbing Brownian paths References Index.

Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions

Abstract: First we define the notion of k-Ricci curvature of a Riemannian n- manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean cur- vature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.

Conformal normal curvature and assessment of local influence

Abstract: In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

Constant scalar curvature metrics with isolated singularities

Abstract: We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8]

A point mass in an isotropic universe: II. Global properties

Abstract: The global structure of McVittie's solution representing a point mass embedded in a spatially flat Robertson-Walker universe is investigated. The scalar curvature singularity at proper radius R=2m, where m (constant) is the Schwarzschild mass, and the apparent horizon which surrounds it are studied. The conformal diagram for the spacetime is obtained via a qualitative analysis of the radial null geodesics. Particular attention is paid to the physical interpretation of this spacetime; previous work on this issue is reviewed, and to how recent quasi-local definitions of black and white holes relate to this spacetime.

Invariant surfaces of the Heisenberg groups

Abstract: We fix a left-invariant metric g in the eisenberg group,H 3, and give a complete classification of the constant mean curvature surfaces (including minimal) which are invariant with respect to 1-dimensional closed subgroups of the connected component of the isometry group of (H 3, g). In addition to finding new examples, we organize in a common framework results that have appeared in various forms in the literature, by the systematic use of Riemannian transformation groups. Using the existence of a family of spherical surfaces for all values of nonzero mean curvature, we show that there are no complete graphs of constant mean curvature. We extend some of these results to the higher dimensional Heisenberg groupsH 2n+1.

On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Abstract: Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kahler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kahler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the case of extremal metrics, the Mabuchi functional is not applicable. Therefore, we need a new (or a modified) functional for metrics which are invariant under a maximal compact connected subgroup K of Aut(M). We did not obtain this functional until the appearing of [FM] (while we were reviewing [FM] in 1995). Mabuchi also found this functional independently [Mb3] (see also [Sm]). A definition of this functional was given in [GC]. We shall give some results and applications in this paper. It turns out that our modified Mabuchi functional M(ω1, ω2) has the property that M(ω1, g∗ω2) = M(ω1, ω2), for any g ∈ CKC(K), where CKC(K) is the centralizer of K in the complexification K of K. Moreover, the extremal metrics are exactly the local minimal points of this functional. Therefore, we expect that the existence of an extremal metric is almost equivalent to the existence of a lower bound of this functional. Surprising enough that the first application of this functional is not the existence but the uniqueness of extremal metrics on smooth toric varieties, i.e., smooth Kahler manifolds with an open (C∗)n-orbit. Therefore, there is for example at most one extremal metric in any Kahler class of the manifold obtained by blowing up two points or three points of a two dimensional complex projective space. To have the uniqueness, we consider the Mabuchi moduli space of the Kahler metrics on the toric varieties (see [Mb2], which was rediscovered by Semmes [Se1] and Donaldson [Ch]). It turns out that the moduli space is flat in this situation (see also [Se1,2]). Moreover, for any two Kahler metrics there is a unique geodesic

Motion by intrinsic Laplacian of curvature

Abstract: In this paper, we discuss numerical schemes to model the motion of curves and surfaces under the intrinsic Laplacian of curvature. This is an intrinsically dif"cult problem, due to the lack of a maximum principle and the delicate nature of computing an equation of motion which includes a fourth derivative term.We design and analyze a host of algorithms to try and follow motion under this #ow, and discuss the virtues and pitfalls of each. Synthesizing the results of these various algorithms, we provide a technique which is stable and handles very delicate motion in two and three dimensions. We apply this algorithm to problems of surface diffusion #ow, which is of value for problems in surface diffusion, metal re#ow in semiconductor manufacturing, sintering, and elastic membrane simulations. In addition, we provide examples of the extension of this technique to anisotropic diffusivity and surface energy which results in an anisotropic form of the equation of motion.

A pointwise inequality in submanifold theory

Abstract: We obtain a pointwise inequality valid for all submanifolds $M^n$ of all real space forms $N^{n+2}(c)$ with $n\ge 2$ and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of $M^n$, and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of $M^n$ in $N^m(c)$.

Kodaira dimension and the Yamabe problem

Abstract: The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M . (To be absolutely precise, one only considers constant-scalar-curvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4-manifold of a complex algebraic surface (M,J), it is shown that the sign of Y (M) is completely determined by the Kodaira dimension Kod(M,J). More precisely, Y (M) 0 iff Kod(M,J) = −∞.

Scalar curvature and hammocks

Abstract: Scalar curvature is the simplest generalization of Gaussian curvature to higher dimensions However there are many questions open with regard to its relation to other geometric quantities and topology Here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature, namely the one-sided affinity for curvature decreasing deformations The first one is concerned with some prescribed decrease of the scalar curvature Scal(g) of some Riemannian metric g on a given manifoldMn of dimension≥ 3 We denote the e−neighborhood of some set U with respect to g by Ue

Translation surfaces with constant mean curvature in 3-dimensional spaces

Abstract: We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E3 and 3-dimensional Minkowski space E 1 3 .

Harnack inequalities on a manifold with positive or negative Ricci curvature

Abstract: Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below

On Einstein Manifolds of Positive Sectional Curvature

Abstract: Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ℂℙ2, with its standard Fubini–Study metric.

The AdS/CFT Correspondence Conjecture and Topological Censorship

Abstract: In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, $\scri$, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result hep-th/9910245, but is independent of the scalar curvature of $\scri$. Furthermore, the topology of V', the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of $\scri$. In this paper, we prove a generalization of the homology results in gr-qc/9902061 in arbitrary dimension, that H_{n-1}(V;Z)=Z^k where V is the closure of V' and k is the number of boundaries $\Sigma_i$ interior to $\Sigma_0$. As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n=2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B^2 or $I\times S^1$.

Scaling solutions in Robertson-Walker spacetimes

Abstract: We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p = (-1) and a non-interacting scalar field with an exponential potential V() = V0e-. We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where 1 ( (2/3),2 3). We find that this matter scaling solution is unstable to curvature perturbations for > (2/3). The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with 2/2 ( > (2/3),2 > 2). We find that solutions with 0 are never late-time attractors.

Optimal pinching constants of odd dimensional homogeneous spaces

Abstract: In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B13=SU(5)/(Sp(2)×ℤ2S1) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W7 1,1=SU(3)/S1 1,1. We prove that the optimal pinching constants are precisely in both cases. So far B13 and W7 1,1 were only known to admit Riemannian metrics with pinching constants.¶We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W7 k,l =SU(3)/S1 k,l . Our computations cover the cone of invariant T2-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.¶In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W7 k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B13 is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ3/ℤ2 and inherits an Sp(2)-invariant Riemannian metric, and the other is W7 1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W7 1,1 and the Berger space B13.