Large deviation and the tangent cone at infinity of a crystal lattice

We discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as we shall observe, has remarkable combinatorial features, and shows up also in the Gromov-Hausdorff limit of a crystal lattice.To the memory of our late friend Robert Brooks     

The tangent cone to a quasimetric space with dilations

We propound some convergence theory for quasimetric spaces that includes as a particular case the Gromov-Hausdorff theory for metric spaces. We prove the existence of the tangent cone (with respect to the introduced convergence) to a quasimetric space with dilations and, as a corollary, to a regular quasimetric Carnot-Carathéodory space. This result gives, in particular, Mitchell’s cone theorem.     

A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with |A|² ≤ 1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.     

Algebraic properties of the tangent cone to a quasimetric space with dilations

Doklady Mathematics -     

Asymptotical flatness and cone structure at infinity

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends, and we classify (except for the case dim M=4, where it remains open if one of the theoretically possible cones can actually arise) for simply connected ends all possible cones at infinity. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat m-manifold with nonnegative sectional curvature is isometric to , whereS is an asymptotically flat surface. Received: 5 January 2000 / Published online: 19 October 2001     

Uniqueness for Ricci flow with unbounded curvature in dimension 2

We consider the uniqueness of Ricci flow with the initial curvature bounded from above, but not necessarily bounded from below, on a 2-dimensional complete noncompact manifold.     

On a characterization of Clarke's tangent cone

For a subsetM of a normed vector spaceX, it is shown that, in the characterization given by Elster and Thierfelder of Clarke's tangent cone toM at a pointx ⁰X, there is a requirement which becomes superfluous whenx ⁰ belongs to the closure ofM.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

Islands at infinity on manifolds of asymptotically nonnegative curvature

We introduce an invariant which measures the R-eccentricity of a point in a complete Riemannian manifold M and show that it goes to zero when the point goes to infinity, if M has asymptotically nonnegative curvature. As a consequence we show that the isometry group is compact if M has asymptotically nonnegative curvature and a point with positive sectional curvature. Both authors were partially supported by CNPq of Brazil and the second author was also partially supported by FAPERJ of Brazil.     

Scattering metrics and geodesic flow at infinity

Any compact ? ^∞ manifold with boundary admits a Riemann metric on its interior taking the form x ⁻⁴ dx ² +x ⁻² h′ near the boundary, where x is a boundary defining function and h′ is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a ‘scattering metric’ was discussed by the first author and here it is shown, as conjectured, that the scattering matrix is a Fourier integral operator which quantizes the geodesic flow on the boundary, for the metric h, at time π. To prove this the Poisson operator, of the associated generalized boundary problem, is constructed as a Fourier integral operator associated to a singular Legendre manifold. Oblatum 24-VII-1995     

Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

We observe that the comparison result of Barles–Biton–Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow. To cite this article: J. Chen, C. Pang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Uniqueness of isometric immersions with the same mean curvature

Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also prove similar results if the mean curvature condition is replaced by an ${\sigma }_2$-curvature condition.     

Decay at infinity of solutions to higher order partial differential equations: Removal of the curvature assumption

In an earlier paper a generalization of Rellich’s theorem on the Helmholz equation was obtained for a large class of higher order equationsP(1/i∂/∂x)u=f, subject to the condition that the Gaussian curvature ofP(ξ)=0 never vanish. This restriction is removed in this note.     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.