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.
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.
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.
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.
In this note, we are concerned with the Kozlowski-Simon conjecture on ovaloids and prove that it is correct under additional conditions.
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.
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...
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....
, 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...
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...