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Self-similarly expanding networks to curve shortening flow

Oliver C. Schnürer, Felix Schulze (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

Soluzioni di tipo barriera

Matteo Novaga (2001)

Bollettino dell'Unione Matematica Italiana

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.

Some aspects of the variational nature of mean curvature flow

Giovanni Bellettini, Luca Mugnai (2008)

Journal of the European Mathematical Society

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 ℱ on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with ℱ . We show some connections between minimizers of ℱ and mean curvature flow.

Some evolution equations under the List's flow and their applications

Bingqing Ma (2014)

Commentationes Mathematicae Universitatis Carolinae

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 L 2 -estimates for generalized scalar curvature and the first variational formulae for non-negative eigenvalues with respect to the Laplacian.

Stability analysis of phase boundary motion by surface diffusion with triple junction

Harald Garcke, Kazuo Ito, Yoshihito Kohsaka (2009)

Banach Center Publications

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 H - 1 -gradient flow structure of the evolution problem and the analysis of eigenvalues of a corresponding differential operator.

Suites de flots de Ricci en dimension 3 et applications

Thomas Richard (2009/2010)

Séminaire de théorie spectrale et géométrie

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...

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