We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.
In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all regular tubular surfaces whose curvatures verify a given polynomial relation $Q(K,H)=0$, and the set $Q(S)$ of all polynomial relations vanished by the curvatures of a given tubular surface $S$.
Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in hyperbolic space.
We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.
This survey introduces to the hyperbolic unfolding correspondence that links the geometric analysis of minimal hypersurfaces with that of Gromov hyperbolic spaces. Problems caused from hypersurface singularities oftentimes become solvable on associated Gromov hyperbolic spaces. Applied to scalar curvature geometry this yields smoothing schemes that eliminate such singularities.
We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.
We construct families of smooth functions $H\colon\mathbb{R}^{n+1}\to\mathbb{R}$ such that the Euclidean $(n+1)$-space is completely filled by not necessarily round hyperspheres of mean curvature $H$ at every point.
Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.
In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fiber compact.
In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\mbox{H}$-hypersurfaces. Then, we give the complete classification of $\mbox{H}$-hypersurfaces with 3 distinct curvatures. We also give explicit examples.
This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature.
In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for submanifolds in hyperbolic spaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial submanifold is positive, but does not imply positivity of the sectional curvature of $M_0$.
We prove that strong finite total curvature complete hypersurfaces of (n+1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punctures.
We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to non degenerate surfaces in Lorentz-Minkowski space.
A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.
In this paper, we consider a class of prescribed Weingarten curvature equations. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the prescribed Weingarten curvature equations.
Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $ε\approx 0$, a closed and embedded curve $u^ε$ in $\mathbb H^2$ having geodesic curvature $k+εK(u^ε)$ at each point.
We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of $Φ$-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature $H$ satisfies $H<1$ under suitable boundary conditions.
The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.
