scholarly journals Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

2009 ◽  
Vol 19 (2) ◽  
pp. 498-519 ◽  
Author(s):  
Alexander Nabutovsky ◽  
Regina Rotman
Keyword(s):  
Riemannian Manifold ◽  
Closed Riemannian Manifold ◽  
Short Geodesic

On the Inner Radius of a Nodal Domain

2008 ◽  
Vol 51 (2) ◽  
pp. 249-260 ◽  
Author(s):  
Dan Mangoubi
Keyword(s):  
Riemannian Manifold ◽  
Lower Bounds ◽  
Type Inequality ◽  
Large Eigenvalue ◽  
Upper And Lower Bounds ◽  
Local Behavior ◽  
Nodal Domain ◽  
Closed Riemannian Manifold ◽  
Inner Radius

AbstractLet M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λα(log λ)β. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincaré type inequality proved by Maz’ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

scholarly journals Focal rigidity of flat tori

2011 ◽  
Vol 83 (4) ◽  
pp. 1149-1158 ◽  
Author(s):  
Ferry Kwakkel ◽  
Marco Martens ◽  
Mauricio Peixoto
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Topological Structure ◽  
Closed Riemannian Manifold ◽  
Flat Tori

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪iΣi called the focal decomposition of TM. The sets Σi are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n > 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

2016 ◽  
Vol 18 (03) ◽  
pp. 1550039 ◽  
Author(s):  
Mónica Clapp ◽  
Marco Ghimenti ◽  
Anna Maria Micheletti
Keyword(s):  
Riemannian Manifold ◽  
Mass Formula ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Perturbation Parameter ◽  
Positive Dimension ◽  
Closed Riemannian Manifold ◽  
Klein Gordon ◽  
Semiclassical States ◽  
Static Form

We establish the existence of semiclassical states for a nonlinear Klein–Gordon–Maxwell–Proca system in static form, with Proca mass [Formula: see text] on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of three-dimensional manifolds on which the system has semiclassical solutions for every exponent [Formula: see text] The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero.

scholarly journals Volumes of balls in Riemannian manifolds and Uryson width

2017 ◽  
Vol 09 (02) ◽  
pp. 195-219 ◽  
Author(s):  
Larry Guth
Keyword(s):  
Riemannian Manifold ◽  
Unit Ball ◽  
Riemannian Manifolds ◽  
Closed Riemannian Manifold ◽  
Small Constant

If [Formula: see text] is a closed Riemannian manifold where every unit ball has volume at most [Formula: see text] (a sufficiently small constant), then the [Formula: see text]-dimensional Uryson width of [Formula: see text] is at most 1.

scholarly journals Convergence of the solutions of discounted Hamilton–Jacobi systems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Davini ◽  
Maxime Zavidovique
Keyword(s):  
Riemannian Manifold ◽  
Coupled System ◽  
Discount Factor ◽  
Limit System ◽  
Specific Solution ◽  
Closed Riemannian Manifold ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Jacobi Equations ◽  
Minimizing Measures

Abstract We consider a weakly coupled system of discounted Hamilton–Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to 0. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton–Jacobi systems and on suitable random representation formulae for the discounted solutions.

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

scholarly journals Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

2017 ◽  
Vol 2017 (731) ◽  
pp. 1-19 ◽  
Author(s):  
Stéphane Sabourau
Keyword(s):  
Riemannian Manifold ◽  
Level Set ◽  
Ricci Curvature ◽  
Morse Function ◽  
Minimal Hypersurface ◽  
Minimal Hypersurfaces ◽  
Nonnegative Ricci Curvature ◽  
Closed Riemannian Manifold

AbstractWe establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

Sign in / Sign up

Export Citation Format

Share Document