Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Comment: 27 pages, 1 figure, TeX, v3. Final version to appear in CMP. Exposition has been extensively tightened and the proof of Proposition 3.5 has been simplifie

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics \sigma_\e collapsing to a subRiemannian metric $\sigma_0$ as \e\to 0. We establish $C^{k,\alpha}$ estimates for this flow, that are uniform as \e\to 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in \cite{CCM3} to the total variation flow.Comment: arXiv admin note: text overlap with arXiv:1212.666

Construction of Maximal Hypersurfaces with Boundary Conditions

We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold

Geodesics in the space of Kähler cone metrics, I

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics Hβ ; that is equivalent to a homogeneous complex Monge–Ampère equation whose boundary values consist of Kähler metrics with cone sin- gularities. Our approach concerns the generalization of the function space defined in Donaldson [25] to the case of Kähler manifolds with boundary; moreover we introduce a subspace HC of Hβ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of Cβ1’1 geodesics whose boundary values lie in HC. Moreover, we prove that such geodesic is the limit of a sequence of Cβ2’α approximate geodesics under the Cβ1’1 -norm. As a geometric application, we prove the metric space structure of HC

Geometry and Topology of some overdetermined elliptic problems

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also some boundedness properties of the solution $u$. Some of our results generalize to higher dimensions