We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.
We present the general theory of barrier solutions in the sense of De Giorgi, and we consider different applications to ordinary and partial differential equations. We discuss, in particular, the case of second order geometric evolutions, where the barrier solutions turn out to be equivalent to the well-known viscosity solutions.
We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with . We show some connections between minimizers of and mean curvature flow.
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.
The linearized stability of stationary solutions for the surface diffusion flow with a triple junction is studied. We derive the second variation of the energy functional under the constraint that the enclosed areas are preserved and show a linearized stability criterion with the help of the -gradient flow structure of the evolution problem and the analysis of eigenvalues of a corresponding differential operator.
Dans cet article, on passe en revue certains résultats dus à Miles Simon sur le flot de Ricci de certains espaces métriques de dimension 3 exposés dans [28] et [26].On commence par voir le lien entre théorèmes de rigidité et convergence des variétés sur un exemple dû à Berger et Durumeric. On remarque ensuite que pour obtenir de tels théorèmes de rigidité en utilisant le flot de Ricci, il faut être capable de construire le flot pour des espaces peu lisses.Les deux dernières partie sont consacrées...