We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem which applies only to calibrated submanifolds of special holonomy ambient manifolds.

Let M be a closed orientable 3-manifold admitting an H2 × R or gSL2(R) geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to S1, according as the center of 1(M) is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings SF(M) for compact orientable aspherical 3-manifolds. We also prove that when M has an H2×R or gSL2(R) geometry and the base orbifold has underlying manifold the 2sphere with three cone points, the inclusion Isom(M) ! Diff(M) is a homotopy equivalence.

Scuola Normale Superiore, Pisa, 1984, tous droits rservs. Laccs aux archives de la revue Annali della Scuola Normale Superiore di Pisa, Classe di Scienze(http://www. sns. it/it/edizioni/riviste/annaliscienze/) implique laccord avec les conditions gnrales dutilisation (http://www. numdam. org/conditions). Toute utilisation commerciale ou impression systmatique est constitutive dune infraction pnale. Toute copie ou impression de ce fichier doit contenir la prsente mention de copyright.

Let  $\sum $ be a compact oriented surface immersed in a four dimensional K¨ahler-Einsteinmanifold $(M, ω)$. We consider the evolution of $\sum$ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When $M$ has two parallel K¨ahler forms $\omega " $ and $\omega ''$ that determine different orientations and  $\sum$ is symplectic with respect to both $\omega " $ and $\omega ''$, we prove the mean curvature flow of $\sum$  exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.