We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots.

H. Hopf showed that the only constant mean curvature sphere S2 immersed in R3 is the round sphere. The K¨ahler framework is an adequate approach to generalize Hopf’ s theorem to higher dimensions. When ϕ : M → Rn is an isometric immersion from a K¨ahler manifold, the complexified second fundamental form α splits according to types. The (1, 1) part of the second fundamental form plays the role of the mean curvature for surfaces and will be called the pluri-mean curvature pmc. Therefore isometric immersions with parallel pluri-mean curvature (ppmc isometric immersions) generalize in a natural way the cmc immersions. It is a standard fact that R8 is the smallest space where CP2 can be embedded. The aim of this work is to generalize Hopf’s theorem proving in particular that the only ppmc isometric immersion from CP2 into R8 is the standard immersion.

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We derive local integral and sup- estimates for the curvature of stable marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical interest, such as dynamical horizons.

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose–type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi–Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on S2.