We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in L ∞ and to the harmonic map flow for initial maps whose image is contained in a small geodesic ball.

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The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Thus our result has applications in the theory of General Relativity

In this paper, we investigate the properties of small surfaces of Willmore type in three-dimensional Riemannian manifolds. By small surfaces, we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball B r ( p) for arbitrarily small radius r around a point p in the Riemannian manifold, then the scalar curvature must have a critical point at p. As a byproduct of our estimates, we obtain a strengthened version of the non-existence result of Mondino [9] that implies the nonexistence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.

Let (M 2 , g) be a compact Riemannian surface and let (N n , h) be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some R l . We consider a sequence u ε ∈ C ∞ (M, N ) (ε → 0) of critical points of the functional E ε (u) = M (|Du| 2 + ε| u| 2 ) with uniformly bounded energy. We show that this sequence converges weakly in W 1,2 (M, N ) and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that N = S l−1 → R l .

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.Comment: Minor correction

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International audienceWe analyze the possible concentration behavior of heat flows relatedto the Moser-Trudinger energy and derive quantization results completelyanalogous to the quantization results for solutions of the corresponding ellipticequation. As an application of our results we obtain the existence of criticalpoints of the Moser-Trudinger energy in a supercritical regim