Showing papers on "Scalar curvature published in 1995"

Constant mean curvature solutions of the Einstein constraint equations on closed manifolds

Abstract: We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.

Fano manifolds, contact structures, and quaternionic geometry

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

The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold.

Abstract: A conjecture of H. Hopf states that if M2n is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic, χ(M2n), should satisfy (-l)nχ(M2n) > 0. In this paper, we investigate the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(O) inequality. In this context, the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. In the case of a manifold with cubical cell structure, the local version is a purely combinatorial statement which can be shown to hold under appropriate conditions. The original conjecture of Hopf, and a similar conjecture for nonnegative curvature (which we shall not be concerned with here), are true in dimensions 2 and 4, by the Chern-Gauss-Bonnet Theorem: in both cases the curvature condition forces the Gauss-Bonnet integrand to have the correct sign. This is immediate in dimension 2. Chern [Ch] gives a proof in dimension 4 and attributes the result to Milnor. A result of [Ge] shows that, in dimensions > 6, the curvature condition does not force the Gauss-Bonnet integrand to have the correct sign; hence, the same argument does not work in higher dimensions. (However, the hypothesis that the curvature operator is negative semidefinite does force the integrand to have the correct sign.)

3-manifolds with(out) metrics of nonpositive curvature

Abstract: In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.

A Computed example of nonuniqueness of mean curvature flow in R3

Abstract: A family of surface (M{sub t}){sub t{element_of}R} in R{sup n} is said to be moving by mean curvature provided. Here H(x) is the mean curvature vector of M{sub t} at x. Is there a smooth hypersurface in some Euclidean space whose mean curvature flow admits nonuniqueness after the onset of singularities? In this note we present compelling numerical evidence for nonuniqueness starting from a certain smooth surface in R{sup 3}. In contrast to other references, we do not have a complete proof for our construction.

Mean curvature flow through singularities for surfaces of rotation

Abstract: In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Geometrical aspects of statistical mechanics

Abstract: Investigation of the geometry of thermodynamic state space, based upon the differential geometric approach to parametric statistics developed by Chentsov [Statistical Decision Rules and Optimal Inference (Nauka, Moscow, 1972)], Efron [Ann. Stat. 3, 1189 (1975)], Amari [Ann. Stat. 10, 357 (1982)], and others, provides a deeper understanding of the mathematical structure of statistical thermodynamics. In the present paper, the Riemannian geometrical approach to statistical mechanical systems due to Janyszek [J. Phys. A 23, 477 (1990)] is applied to various models including the van der Waals gas and magnetic models. The scalar curvature for these models is shown to diverge not only at the critical points but also along the entire spinodal curve. The critical behavior of the curvature derived from the Fisher information metric turns out to coincide with that derived from the entropy differential metric by Ruppeiner [Phys. Rev. A 20, 1608 (1979)].

Constant mean curvature surfaces constructed by fusing Wente tori

Abstract: Closed smooth surfaces of any genus g ? 2, immersed in E3 and of constant mean curvature, are constructed, by "fusing" Wente tori. A fundamental problem in mathematics whose answer has important consequences in many scientific disciplines is the so-called isoperimetric problem, which asks which surface has the least area among the surfaces enclosing a fixed volume. The answer is well known to be the round sphere, although a rigorous proof of this fact requires a certain amount of mathematical ingenuity and sophistication. More generally, one can ask which surfaces have critical area subject to the requirement that they enclose a fixed volume. Such surfaces are often called soap bubbles because a soap film-or more generally a fluid interface-in equilibrium between two areas of different pressure and subject only to the forces induced by this pressure and the surface tension has critical area subject to a fixed enclosed volume constraint. The differential equation characterizing such surfaces locally is H = constant $ 0, where H is the mean curvature. We adopt the abbreviations from now on "CMC surface" to stand for "nonzero constant mean curvature, immersed, smooth, closed surface in E3," and "CMC immersion" for the corresponding immersion. E3 stands for Euclidean threedimensional space (equipped with the usual flat metric), a mathematical model of the space we live in. Since closed surfaces have no minimal immersions into E3, a CMC immersion is from the geometric point of view most appealing among all E3 immersions of the surface. One is led then to the question of which surfaces have CMC immersions and more specifically whether there are any CMC surfaces that are not round spheres. Such questions have been studied seriously by differential geometers for a long time, and in 1853 Jellet proved that star-shaped CMC surfaces are round spheres (1). A century later Hopf established the same for topological CMC spheres (2), andshortly afterwards Alexandrov did the same for embedded CMC surfaces (3). Finally, Barbosa and doCarmo showed that local minimizers of the variational problem are round spheres (4). Because of these results the prevailing mood for a while was that the round spheres are the only CMC surfaces. In 1982 Hsiang demonstrated, however, that this fails, at least in higher dimensions (5). In 1984 Wente in a surprising development achieved the construction of nonspherical CMC surfaces (6). The first examples were tori and the methods used made use of the facts that the Hopf differential, a quadratic holomorphic differential associated to a CMC surface, takes a particularly simple form on a torus and that the commutativity of the fundamental group of the torus forces all of its representations into the Euclidean group to consist of motions sharing a common axis. The Wente construction has been analyzed and extended (7-12), and today we have a classification and a much better understanding of the CMC tori. These methods, however, have failed to The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. ?1734 solely to indicate this fact. produce any results up to now for CMC surfaces of genus g ?2. In 1987 the author announced the construction of CMC surfaces of any genus g 2 3 (13). The methods used were entirely different from the above, and the construction consisted of fusing Delaunay surfaces (14) to obtain surfaces of mean curvature H close to 1, then perturbing the latter to get the desired surfaces with H 1 (13, 15, 16). Delaunay surfaces can be parametrized by a parameter T so that a fundamental domain of the surface consists of the following two pieces: First, a positively curved region that is a perturbation of a round sphere minus two diametrically opposed small (as T -O 0) discs. The perturbation is of order 1X1 on compact sets of the round sphere minus the appropriate antipodal points. Second, a negatively curved region which expanded by a factor 1/h tends to a catenoid as T -O 0. The circles where the curvature changes sign are of order 'P. The fusion involves the spherical pieces. The construction works provided that a certain balancing condition is satisfied at each nearly spherical region where fusion occurs. The CMC surfaces constructed this way are actually Z-coverings of the closed surface, where the period involved is a translation which can be arranged to vanish by appropriately varying the T parameter (16). The balancing condition turns out to prohibit the construction of genus g = 2 CMC surfaces by this construction. The author hopes that the construction above can be generalized to a construction where CMC surfaces-including hypersurfaces in En (n 2 3) besides the usual case of surfaces in E3 (notice that there is nothing intrinsically two-dimensional in this method)-are constructed by using as ingredients complete minimal (hyper)surfaces scaled to be of very small size and with a neighborhood of their infinity removed and round spheres with a number of small discs removed. If these ingredients are appropriately placed and of the appropriate size, one could join them by appropriate surfaces so that a closed surface of mean curvature H close to 1 would be obtained and subsequently perturbed to a CMC surface. There are many difficult obstacles to be overcome before such a general construction can be carried out. Specifying correctly the joining pieces in particular seems especially difficult. The purpose of this paper is to announce a much more modest generalization of the old construction: The old construction is extended by allowing the fusion of Wente tori as well-such a fusion is possible because the Wente tori, like the Delaunay cylinders, contain nearly round-spherical regions. This way the joining pieces above are "borrowed" from the Wente tori, as in the old construction they were "borrowed" from the Delaunay cylinders. The new construction is enough to settle the question of whether all surfaces have CMC immersions, because it allows us to "fuse" g Wente tori for any g 2 2. It turns out that to prove that this construction works, we need a much deeper understanding of the mathematical Abbreviations: CMC, constant mean curvature; asr, almost spherical region.

On the Conformal and CR Automorphism Groups

Abstract: This paper concerns the behavior of conformal diffeomorphisms between Riemannian manifolds, and that of CR diffeomorphisms between strictly pseudoconvex CR manifolds, We develop a new approach to the subject which involves the scalar curvature theory, and the conformally invariant Laplace operator, and its subelliptic analogue in the CR case. We first describe the results in the conformal case. Our main quantitative result (Proposition 2.1) concerns a conformal embedding of a ball with a well controlled Riemannian metric into a Riemannian manifold with zero scalar curvature. It says that if the derivative of the transformation is large at the origin, then it is uniformly large, so that the map is uniformly expanding. Furthermore, it shows that the full curvature tensor on the image of the ball must be very small. This result can be immediately applied to show that the conformal automorphism group G of a scalar flat manifold (not necessarily complete) acts properly unless the manifold is isometric to R n .

A necessary and sufficient condition for the nirenberg problem

Abstract: We seek metrics conformal to the standard ones on Sn having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S2, for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

Abstract: We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.

Spaces with Curvature Bounded Below

Abstract: After the seminal work of Gromov (see [G1],[GLP]), questions of this type, with various assumptions on curvatures, and other geometric characteristics, have been receiving much attention. Cheeger, Gromov, and Fulraya developed a farreaching theory of collapse under the two-sided curvature bounds (see [CFG]). One of the simplest conclusions of this theory is that the limit space M in this case has a stratification, each stratum being a totally geodesic Riemannian manifold. (This stratification is nontrivial unless Mj admits a structure of locally trivial fibration over M.) This conclusion can be explained by the following argument (see [GLP, 8.30] for details). Suppose that Mj have sectional curvatures between —1 and 1, and let pj G Mj converge to some point p G M. Then the balls Bj of radius 7r/2 centered at pj are covered by convex balls Bj of the same radius in the tangent spaces at pj, endowed with the lifted metrics. The sequence Bj has the same curvature bounds and, in addition, a uniform lower bound on the injectivity radius; therefore its limit B is a manifold. Now each Bj is a quotient of Bj by an isometric (pseudo)group action, hence a ball B in M centered at p must be a quotient of B by the limit action; this leads to a stratification of B with totally geodesic strata.

Connected sum constructions for constant scalar curvature metrics

Abstract: We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making "analytic" connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's [S1] well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.

Polarized 4-Manifolds, Extremal Kahler Metrics, and Seiberg-Witten Theory

Abstract: Using Seiberg-Witten theory, it is shown that any Kahler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L 2 -norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H 2 (M) = H + � H. This implies, for ex- ample, that any such metric on a minimal ruled surface must be locally symmetric.

Curvature corrections to dynamics of domain walls.

Abstract: The most usual procedure for deriving curvature corrections to effective actions for topological defects is subjected to a critical reappraisal. A logically unjustified step (leading to overdetermination) is identified and rectified, taking the standard domain wall case as an illustrative example. Using the appropriately corrected procedure, we obtain a new exact (analytic) expression for the corresponding effective action contribution of quadratic order in the wall width, in terms of the intrinsic Ricci scalar $R$ and the extrinsic curvature scalar $K$. The result is proportional to $cK^2-R$ with the coefficient given by $c\simeq 2$. The resulting form of the ensuing dynamical equations is obtained in terms of the second fundamental form and the Dalembertian of its trace, K. It is argued that this does not invalidate the physical conclusions obtained from the "zero rigidity" ansatz $c=0$ used in previous work.

Geodesic Flows on Manifolds of Negative Curvature

Abstract: This text is based on the lectures given in the Summer school on Dynamical Systems in Trieste, June, 1992. The main motivation was to expose one of the most beautiful and classical chapters of ergodic theory using some basic achievements in the entropy theory of dynamical systems. Another reason was more pragmatic. The interest to geodesic flows on manifolds of negative curvature grew enormously during the last years due to the development of quantum class. A lot of numerical and qualitative facts discovered have mainly by physicists suggest difficult and important problems concerning the connection of eigen-values of Laplacians on compact manifolds of negative curvature and geodesics, especially closed geodesics on such manifolds. We believe that the theory which is explained below can be useful for attacking these problems.

Positive Scalar Curvature Metrics — Existence and Classification Questions

Abstract: Let M be an n-dimensional manifold (all manifolds considered in this paper are smooth, compact, and, unless otherwise specified, their boundary is empty).

On the scalar curvature of complex surfaces

Abstract: Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This extends previous results of Witten and Kronheimer.

The Nirenberg problem in a domain with boundary

Abstract: There has been much work on the Nirenberg problem: which function K(x) on S is the scalar curvature of a metric g on S pointwise conformal to the standard metric g0? It is quite natural to ask the following question on the half sphere S−: which function K(x) on S− is the scalar curvature of a metric g on S− which is pointwise conformal to the standard metric g0 with ∂S− being minimal with respect to g? For n = 2, this has been studied by J. Q. Liu and P. L. Li in [LL]. In this note we study the higher dimensional cases along the lines of [L1-2]. For much work on the Nirenberg problem see, for example, [L1-2] and the references therein. See also some more recent work in [CL1], [HL], [Bi1-2], [SZ], [B], [ChL] and [CL2]. For n ≥ 3, by writing g = ug0, the problem is equivalent to solving the following Neumann problem on S− = {(x1, . . . , xn+1) ∈ S | xn+1 < 0}:

Convergence of a finite element method for non-parametric mean curvature flow

Abstract: Convergence for the spatial discretization by linear finite elements of the non-parametric mean curvature flow is proved under natural regularity assumptions on the continuous solution. Asymptotic convergence is also obtained for the time derivative which is proportional to mean curvature. An existence result for the continuous problem in adequate spaces is included.

La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes

Abstract: K.D. Kirchberg [Ki1] gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact Kahler manifoldM of odd complex dimension with positive scalar curvature. We prove that manifolds of real dimension 8l+6 satisfying the limiting case are twistor space (cf. [Sa]) of quaternionic Kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l+2 satisfying the limiting case is the complex projective spaceCP4l+1.