### A combinatorial curvature flow for compact 3-manifolds with boundary.

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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...

We describe a new link between Perelman’s monotonicity formula for the reduced volume and ideas from optimal transport theory.

In this paper we consider Riemannian manifolds $({M}^{n},g)$ of dimension $n\ge 5$, with semi-positive $Q$-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 $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show...

Let us consider a projective manifold ${M}^{n}$ and a smooth volume form $\Omega $ on $M$. We define the gradient flow associated to the problem of $\Omega $-balanced metrics in the quantum formalism, the $\Omega $-balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the $\Omega $-balancing flow converges towards a natural flow in Kähler geometry, the $\Omega $-Kähler flow. We also prove the long time existence of the $\Omega $-Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...

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 $\epsilon $, and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...

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...

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 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:...

On résume les proprietés de l’invariant $\overline{\lambda}$ de Perelman, et en combinaison avec l’invariant de Yamabe on exprime certaines proprietés géométriques des variétés de dimension $3$ en fonction de $\overline{\lambda}$. On décrit des exemples d’annulation de $\overline{\lambda}$ en dimension $4$, 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 $\overline{\lambda}$ sur les variétés de courbure négative ou nulle de dimension...

We generalize brownian motion on a riemannian manifold to the case of a family of metrics which depends on time. Such questions are natural for equations like the heat equation with respect to time dependent laplacians (inhomogeneous diffusions). In this paper we are in particular interested in the Ricci flow which provides an intrinsic family of time dependent metrics. We give a notion of parallel transport along this brownian motion, and establish a generalization of the Dohrn–Guerra or damped...

In this paper we have studied conformal curvature tensor, conharmonic curvature tensor, projective curvature tensor in Lorentzian $\alpha $-Sasakian manifolds admitting conformal Ricci soliton. We have found that a Weyl conformally semi symmetric Lorentzian $\alpha $-Sasakian manifold admitting conformal Ricci soliton is $\eta $-Einstein manifold. We have also studied conharmonically Ricci symmetric Lorentzian $\alpha $-Sasakian manifold admitting conformal Ricci soliton. Similarly we have proved that a Lorentzian $\alpha $-Sasakian...

We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.