Some applications of Ricci flow to 3-manifolds
Some conditions for a surface in to be a part of the sphere
In this paper some properties of an immersion of two-dimensional surface with boundary into are studied. The main tool is the maximal principle property of a solution of the elliptic system of partial differential equations. Some conditions for a surface to be a part of a 2-dimensional spheren in are presented.
Some evolution equations under the List's flow and their applications
In this paper, we consider some evolution equations of generalized Ricci curvature and generalized scalar curvature under the List’s flow. As applications, we obtain -estimates for generalized scalar curvature and the first variational formulae for non-negative eigenvalues with respect to the Laplacian.
Some existence results for the scalar curvature problem via Morse theory
We prove existence of positive solutions for the equation on , arising in the prescribed scalar curvature problem. is the Laplace-Beltrami operator on , is the critical Sobolev exponent, and is a small parameter. The problem can be reduced to a finite dimensional study which is performed with Morse theory.
Some non-linear function theoretic properties of Riemannian manifolds.
We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.
Some progress in conformal geometry.
Some remarks on the Kozlowski-Simon conjecture for affine ovaloids
In this note, we are concerned with the Kozlowski-Simon conjecture on ovaloids and prove that it is correct under additional conditions.
Some remarks on the weak maximum principle.
We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.
Some topics in spectral geometry
Spaces with Ricci curvature bounds.
Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces
The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that...
Spectral isolation of bi-invariant metrics on compact Lie groups
We show that a bi-invariant metric on a compact connected Lie group is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric on there is a positive integer such that, within a neighborhood of in the class of left-invariant metrics of at most the same volume, is uniquely determined by the first distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where is simple, can be chosen to be two....
Spectre conforme et métriques extrémales
Stability problems in a theorem of F. Schur.
Structure of second-order symmetric Lorentzian manifolds
, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor , are characterized by several geometric properties, and explicitly presented. Locally, they are a product where each factor is uniquely determined as follows: is a Riemannian symmetric space and is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., at some point), the curvature tensor turns out to...
Sufficient decrease principle on Riemannian manifolds.
Sur le problème inverse du calcul des variations : existence de lagrangiens associés à un spray dans le cas isotrope
En utilisant la version de Spencer-Goldschmidt du théorème de Cartan-Kähler nous étudions les conditions nécessaires et suffisantes pour qu’un système d’équations différentielles ordinaires du second ordre soit le système d’Euler-Lagrange associé à un lagrangien régulier. Dans la thèse de Z. Muzsnay cette technique a été déjà appliquée pour donner une version moderne du papier classique de Douglas qui traite le cas de la dimension 2. Ici nous considérons le cas où la dimension est arbitraire, nous...
Sur le volume minimal des variétés ouvertes
Surfaces à courbure moyenne constante et inégalité de Wente
The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces.