Related papers: Complete negatively curved immersed ends in $\Bbb …

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end…

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our…

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an…

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S^3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a…

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact…

We extend recent results of Guan and Spruck, proving existence results for constant Gaussian curvature hypersurfaces in Hadamard manifolds.

The main goal of this paper is to show a counterexample to the following conjecture: {\bf Conjecture} [Meeks, Sullivan]: If $f:M\to \mathbb{R}^3$ is a complete proper minimal immersion where $M$ is a Riemannian surface without boundary and…

We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N(S,H)$ with…

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$,…

We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into $\mathbb{R}^3$. The main tool we use is the relative entropy method employed in the setting of…

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to…

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and…

In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such…

We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,\alpha}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)<…

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has…

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes…

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance.…