Curvature bounds for surfaces in hyperbolic 3-manifolds

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.Comment: 21 pages, 9 figures, published version, added figures, fixed typo

Cohomogeneity-one G2-structures

G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein of positive scalar curvature and are uniquely determined by the simple symmetry group. During the proof the equations for G2-symplectic and G2-cosymplectic structures are studied and the topological types of the manifolds admitting such structures are determined. New examples of compact G2-cosymplectic manifolds and complete G2-symplectic structures are found.Comment: 23 page

Minimal immersions of closed surfaces in hyperbolic three-manifolds

We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(\sigma, t\alpha)$, where $\sigma$ is a conformal structure on a topological surface $S$, and $\alpha dz^2$ is a holomorphic quadratic differential on the surface $(S,\sigma)$. We show that, for each $t \in (0,\tau_0)$ for some $\tau_0 > 0$, depending only on $(\sigma, \alpha)$, there are at least two minimal immersions of closed surface of prescribed second fundamental form $Re(t\alpha)$ in the conformal structure $\sigma$. Moreover, for $t$ sufficiently large, there exists no such minimal immersion. Asymptotically, as $t \to 0$, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.Comment: 16 page

Generalized Reduction Procedure: Symplectic and Poisson Formalism

We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting.Comment: 39 pages, Latex file, Vienna ESI 28 (1993

A rigidity theorem for nonvacuum initial data

In this note we prove a theorem on non-vacuum initial data for general relativity. The result presents a ``rigidity phenomenon'' for the extrinsic curvature, caused by the non-positive scalar curvature. More precisely, we state that in the case of asymptotically flat non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported.Comment: This is an extended and published version: LaTex, 10 pages, no figure

Boundary clustered layers near the higher critical exponents

We consider the supercritical problem {equation*} -\Delta u=|u| ^{p-2}u\text{\in}\Omega,\quad u=0\text{\on}\partial\Omega, {equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\ast}:=\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\mathbb{R}^{N-k})$ in $L^{q}(\mathbb{R}^{N-k})$, $1\leq k\leq N-3.$ We show that in some suitable domains $\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\partial\Omega$ as $p$ approaches $2_{N,k}^{\ast}$ from below. Key words:Nonlinear elliptic boundary value problem; critical and supercritical exponents; existence of positive and sign changing solutions

On strong unique continuation of coupled Einstein metrics

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite order around a point, then they must coincide, up to a local diffeomorphism, in a neighborhood of the point. The novelty of our method lies in the use of a Carleman inequality and thus circumventing the use of analyticity; thus the method is robust under certain non-analytic perturbations. As an example, we also show the strong unique continuation property for the Riemannian Einstein-scalar-field system with cosmological constant.Comment: 12 pages; some minor errors are fixed in revision, some clarifications are mad

On the Riemannian geometry of Seiberg-Witten moduli spaces

We construct natural Riemannian metrics on Seiberg-Witten moduli spaces and study their geometry