A combinatorial curvature flow for compact 3-manifolds with boundary.
A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds
We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...
A logarithmic Gauss curvature flow and the Minkowski problem
A note on Ricci flow and optimal transportation
We describe a new link between Perelman’s monotonicity formula for the reduced volume and ideas from optimal transport theory.
A note on secantoptics.
A regularity theorem for curvature flows.
A singular example for the averaged mean curvature flow.
A strong maximum principle for the Paneitz operator and a non-local flow for the -curvature
In this paper we consider Riemannian manifolds of dimension , with semi-positive -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive -curvature. Modifying the test function construction of Esposito-Robert, we show...
About the Calabi problem: a finite-dimensional approach
Let us consider a projective manifold and a smooth volume form on . We define the gradient flow associated to the problem of -balanced metrics in the quantum formalism, the -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the -balancing flow converges towards a natural flow in Kähler geometry, the -Kähler flow. We also prove the long time existence of the -Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...
Analysis of a semidiscrete scheme for solving image smoothing equation of mean curvature flow type.
Analysis of total variation flow and its finite element approximations
We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter , and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...
Analysis of total variation flow and its finite element approximations
We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since...
Anisotropic geometric functionals and gradient flows
We survey some recent results on the gradient flow of an anisotropic surface energy, the integrand of which is one-homogeneous in the normal vector. We discuss the reasons for assuming convexity of the anisotropy, and we review some known results in the smooth, mixed and crystalline case. In particular, we recall the notion of calibrability and the related facet-breaking phenomenon. Minimal barriers as weak solutions to the gradient flow in case of nonsmooth anisotropies are proposed. Furthermore,...
Asymptotic behavior of solutions to an area-preserving motion by crystalline curvature
Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex:...
Atoroïdalité complète et annulation de l’invariant de Perelman
On résume les proprietés de l’invariant de Perelman, et en combinaison avec l’invariant de Yamabe on exprime certaines proprietés géométriques des variétés de dimension en fonction de . On décrit des exemples d’annulation de en dimension , où on trouve des liens entre l’effondrement et l’existence de métriques à courbure scalaire positive. On montre qu’une version d’atoroïdalité qu’on appelle atoroïdalité complète est détectée par sur les variétés de courbure négative ou nulle de dimension...