Abstract. In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle α for the corresponding Lagrangian submanifold in the cross product space L × M satisfies osc(α) ≤ π. If one considers a 4-dimensional Kähler-Einstein manifold M of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that L ⊂ M is a compact oriented Lagrangian submanifold w.r.t. J such that the Kähler form κ w.r.t. K restricted to L is positive and osc(α) ≤ π, then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. κ.

In this paper, we introduce the Sasaki-Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki-Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler-Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler-Ricci flow.

We consider the flow of a strictly convex hypersurface driven by the Gauß curvature. For the Neumann boundary value problem and for the second boundary value problem we show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauß curvature equation. We also discuss oblique boundary value problems and flows for Hessian equations.  2003 Éditions scientifiques et médicales Elsevier SAS MSC: primary 53C44; secondary 35K20, 53C42 RÉSUMÉ.-Nous considérons le flot d'une hypersurface strictement convexe piloté par la courbure de Gauß. Pour le problème aux limites de Neumann et pour le deuxième problème aux limites nous montrons qu'un tel flot existe pour tout temps et converge vers une solution de l'équation de courbure de Gauß prescrite. Nous étudions aussi des problèmes aux limites oblique et les flots pour des équations hessiennes.  2003 Éditions scientifiques et médicales Elsevier SAS

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T 2n is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space N , where N equals IR × N equipped with a warped product metric and show that a submanifold in N converges to a conformal soliton under the mean curvature flow in N if and only if its associated submanifold in N converges to a minimal submanifold under a rescaled mean curvature flow in N . We then define a notion of stability for conformal solitons and obtain L p estimates as well as pointwise estimates for the curvature of stable solitons.

Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform C 2,α -bounds in space and C 2 -estimates in time for the underlying Monge-Ampère equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result. (2000): 53C44 Mathematics Subject Classification

In this text we outline the major techniques, concepts and results in mean curvature flow with a focus on higher codimension. In addition we include a few novel results and some material that cannot be found elsewhere. Mean curvature flowMean curvature flow is perhaps the most important geometric evolution equation of submanifolds in Riemannian manifolds. Intuitively, a family of smooth submanifolds evolves under mean curvature flow, if the velocity at each point of the submanifold is given by the mean curvature vector at that point. For example, round spheres in euclidean space evolve under mean curvature flow while concentrically shrinking inward until they collapse in finite time to a single point, the common center of the spheres. Mullins [Mul56] proposed mean curvature flow to model the formation of grain boundaries in annealing metals. Later the evolution of submanifolds by their mean curvature has been studied by Brakke [Bra78] from the viewpoint of geometric measure theory. Among the first authors who studied the corresponding nonparametric problem were Temam [Tem76] in the late 1970's and Gerhardt [Ger80] and Ecker [Eck82] in the early 1980's. Pioneering work was done by Gage [Gag84], Gage & Hamilton [GH86] and Grayson [Gra87] who proved that the curve shortening flow (more precisely, the "mean" curvature flow of curves in R 2 ) shrinks embedded closed curves to "round" points. In his seminal paper Huisken [Hui84] proved that closed convex hypersurfaces in euclidean space R m+1 , m > 1 contract to single round points in finite time (later he extended his result to hypersurfaces in Riemannian manifolds that satisfy a suitable stronger convexity, see [Hui86]). Then, until the mid 1990's, most authors who studied mean curvature flow mainly considered hypersurfaces, both in euclidean and Riemannian manifolds, whereas mean curvature flow in higher codimension did not play a great role. There are various reasons for this, one of them is certainly the much different geometric situation of submanifolds in higher codimension since the normal bundle and the second fundamental tensor