Complete Riemannian Metric Research Articles

Complete Riemannian metrics with holonomy group G 2 on deformations of cones over S 3 × S 3

Complete Riemannian metrics with holonomy group G2 on manifolds obtained by deformation of cones over S3 × S3 are constructed.

Convergence of Nonlinear Observers on $\BBR^{n}$ With a Riemannian Metric (Part I)

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally negative. Moreover, we establish that the existence of this metric is related to the observability of the system's linearization along its solutions. Moreover, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. Conversely, we establish that, if a complete Riemannian metric has a Lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property, then there exists an observer with an infinite gain margin.

Completeness in Supergravity Constructions

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component $${\mathcal{H}}$$ of a hypersurface {h = 1} defined by a homogeneous cubic polynomial h such that $${-\partial^2h}$$ is a complete Riemannian metric on $${\mathcal{H}}$$ defines a complete projective special Kähler manifold and any complete projective special Kähler manifold defines a complete quaternionic Kähler manifold of negative scalar curvature. We classify all complete quaternionic Kähler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.

Constant [formula omitted]-curvature metrics with Delaunay type ends

In this paper we produce families of complete noncompact Riemannian metrics with positive constant σk-curvature equal to 2−k(nk) by performing the connected sum of a finite number of given n-dimensional Delaunay type solutions, provided 2⩽2k<n. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S 3. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Moduli spaces of nonnegative sectional curvature and non-unique souls

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnega- tive sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

Complex hyperbolic hyperplane complements

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n − 1)-submanifold S. The main result is that the fundamental group of $${M{\setminus} S}$$ is relatively hyperbolic, relative to fundamental groups of the ends of $${M{\setminus} S}$$ , and $${M{\setminus} S}$$ admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n > 1 the fundamental group of $${M{\setminus} S}$$ satisfies Mostow-type Rigidity, has solvable word and conjugacy problems, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of $${M{\setminus} S}$$ is biautomatic and satisfies Strong Tits Alternative.

Ricci flow on open 3–manifolds and positive scalar curvature

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1 or to some mem- ber of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a variant of Perelman's surgery construction for Ricci flow.

Structure of Whittaker groups and applications to conformal involutions on handlebodies

The geometrically finite complete hyperbolic Riemannian metrics in the interior of a handlebody of genus g, having injectivity radius bounded away from zero, are exactly those produced by Schottky groups of rank g; these are called Schottky structures. A Whittaker group of rank g is by definition a Kleinian group K containing, as an index two subgroup, a Schottky group Γ of rank g. In this case, K corresponds exactly to a conformal involution on the handlebody with Schottky structure given by Γ. In this paper we provide a structural description of Whittaker groups and, as a consequence of this, we obtain some facts concerning conformal involutions on handlebodies. For instance, we give a formula to count the type and the number of connected components of the set of fixed points of a conformal involution of a handlebody with a Schottky structure in terms of a group of automorphisms containing the conformal involution.

Infinitesimal harmonic transformations and ricci solitons on complete Riemannian manifolds

Ricci solitons were introduced by R. Hamilton as natural generalizations of Einstein metrics. A Ricci soliton on a smooth manifold M is a triple (g0,ξ, λ), where g0 is a complete Riemannian metric, ξ a vector field, and λ a constant such that the Ricci tensor Ric0 of the metric g0 satisfies the equation O2 Ric0 = Lξg0 + 2λgo. The following statement is one of the main results of the paper. Let (g0,ξ, λ) be a Ricci soliton such that M,g0 is a complete noncompact oriented Riemannian manifold, $$ \int\limits_M {\left\| \xi \right\|dv < \infty } $$ , and the scalar curvature s0 of g0 has a constant sign on M, then (M, g0) is an Einstein manifold

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples.

Convexity in locally conformally flat manifolds with boundary

of constant positive scalar curvature R(g) = n(n 1) and conformally related to the Euclidean metric . We will also assume that g has nonnegative boundary mean curvature. Here, and throughout this paper, second fundamental forms will be computed with respect to the inward unit normal vector. In this talk we prove Theorem 0.1. If B B1 is a standard Euclidean ball, then @B is convex with respect to the metric g. This theorem is motivated by an analogous one on the sphere due to R. Schoen [3]. He shows that if S n n 3, is closed and nonempty and g is a complete Riemannian metric on S n , conformal to the standard round metric g0 and with constant positive scalar curvature n(n 1), then every standard ball B S n is convex with respect to the metric g. Schoen used this geometrical result to prove the compactness of the set of solutions to the Yamabe problem in the locally conformally flat case. Later, D. Pollack also used Schoen’s theorem to prove a compactness result for the singular Yamabe problem on the sphere where the singular set is a finite collection of points

G 2-holonomy metrics connected with a 3-Sasakian manifold

We construct complete noncompact Riemannian metrics with G 2-holonomy on noncompact orbifolds that are ℝ3-bundles with the twistor space as a spherical fiber.

Large-scale conformal rigidity in dimension three

Let X be a noncompact smooth manifold. Two complete Riemannian metrics on X in a given conformal class can have very different asymptotic geometries. For instance, starting with the Euclidean plane E2 with polar coordinates (r, θ), multiplying the Euclidean metric g0 = dr2 + r2dθ2 by a function equal to 1/r2 outside a compact neighborhood of the origin, one obtains a complete Riemannian metric g which is quasi-isometric to a half-line. The metrics g and g0 are in the same conformal class, but they are not quasi-isometric. In fact, they have different asymptotic dimensions. One can ask whether something similar can happen to two finitely generated groups and 0: can they be ‘coarsely quasi-conformal’ in some sense and yet not quasi-isometric? To give a precise meaning to this question, we will choose Riemannian manifoldsX0,X that are geometric models for 0, and consider conformal mappings between X0 and X. First let us recall some definitions. Two metric spaces (X1,d1) and (X2,d2) are quasi-isometric if there are constants λ ≥ 1 and C ≥ 0 and a map f : X1 → X2 satisfying: λ−1 d1(x, x′) −C ≤ d2(f (x), f (x′)) ≤ λd1(x, x′)+ C ∀ x, x′ ∈ X1 ∀ y ∈ X2, ∃x ∈ X1, d(f (x), y) ≤ C.

Curvature and rank of Teichmüller space

Let S be a surface with genus g and n boundary components, and let d(S) = 3 g - 3 + n denote the number of curves in any pants decomposition of S . We employ metric properties of the graph of pants decompositions P ( S ) to prove that the Weil-Petersson metric on Teichmüller space Teich ( S ) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] k , k ≥ 2); when d(S) ≤ 2, we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of P ( S ) to prove Gromov-hyperbolicity. We prove moreover that Teich ( S ) admits no geodesically complete, Mod ( S )-invariant, Gromov-hyperbolic metric of finite covolume when d(S) ≥ 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) ≥ 2.

Necessary and sufficient conditions for global‐in‐time existence of solutions of ordinary, stochastic, and parabolic differential equations

We derive necessary and sufficient conditions for global-in-time existence of solutions of ordinary differential, stochastic differential, and parabolic equations. The conditions are formulated in terms of complete Riemannian metrics on extended phase spaces (conditions with two-sided estimates) or in terms of derivatives of proper functions on extended phase spaces (conditions with one-sided estimates).

Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

Let M be a complete Riemannian metric of sectional curvature within [−a 2,−1] whose fundamental group contains a k-step nilpotent subgroup of finite index. We prove that a ≥ k answering a question of M. Gromov. Furthermore, we show that for any $$\epsilon > 0,$$ the manifold M admits a complete Riemannian metric of sectional curvature within $$[ - (k + \epsilon )^2 , - 1].$$

Proper smooth G-manifolds have complete G-invariant Riemannian metrics

Assume G is a Lie group, K is a compact subgroup of G and M is a proper smooth G-manifold. Using properties of the regular representations L 2 ( G ) and L 2 ( K ) , we first prove results about extending certain representations and embedding homogeneous spaces smoothly into Hilbert G-spaces. We then prove that M can be embedded as a closed smooth G-invariant submanifold of some Hilbert G-space. It follows that M admits a complete G-invariant smooth Riemannian metric.

Globally hyperbolic flat space–times

We consider (flat) Cauchy-complete GH space–times, i.e., globally hyperbolic flat Lorentzian manifolds admitting some Cauchy hypersurface on which the ambient Lorentzian metric restricts as a complete Riemannian metric. We define a family of such space–times—model space–times—including four subfamilies: translation space–times, Misner space–times, unipotent space–times, and Cauchy-hyperbolic space–times (the last family—undoubtful the most interesting one—is a generalization of standard space–times defined by G. Mess). We prove that, up to finite coverings and (twisted) products by Euclidean linear spaces, any Cauchy-complete GH space–time can be isometrically embedded in a model space–time, or in a twisted product of a Cauchy-hyperbolic space–time by flat Euclidean torus. We obtain as a corollary the classification of maximal GH space–times admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model space–time.