Eigenvalue estimates for minimal surfaces in hyperbolic space

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.

We present a Gauss–Bonnet type formula for complete surfaces in n-dimensional hyperbolic space $$\mathbb{H}^{n}$$ under some assumptions on their asymptotic behaviour. As in recent results for Euclidean submanifolds (see Dillen–Kühnel [4] and Dutertre [5]), the formula involves an ideal defect, i.e., a term involving the geometry of the set of points at infinity.

A surface in hyperbolic space H 3 invariant by a group of parabolic isometries is called a parabolic surface.In this paper we investigate parabolic surfaces of H 3 that satisfy a linear Weingarten relation of the form aκ 1 +bκ 2 = c or aH +bK = c, where a, b, c ∈ R and, as usual, κ i are the principal curvatures, H is the mean curvature and K is de Gaussian curvature.We classify all parabolic linear Weingarten surfaces in hyperbolic space.

In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Osserman type inequality for such surfaces was shown. Though its equality condition is not solved yet, the authors have noticed that the equality condition of the original Chern-Osserman inequality itself is not found in any literature except for the case n=3, in spite of its importance. In this paper, a simple geometric condition for minimal surfaces that attains equality in the Chern-Osserman inequality is given. The authors hope it will be a useful reference for readers.

The first broad line of demarcationi is naturally between those surfaces which have a vanishing discriminant, and those which have a non-vanishing one. We shall occupy ourselves only with surfaces of the latter type. The next great division is into ruled surfaces, non-ruled ones, and surfaces whose equations are definite forms. This classification depends upon the signs of the discriminant and its leading minors, according to certain familiar principles. We then come to the metrical properties of quadries. These arise from the various possible relations to the Absolute, so that the next problem is that of the classification of the mutual relations of two quadrics. This has been successfully solved by a number of writers, in particular, by HESSE t and CLEBSCH.1 Yet from the strict point of view of a dweller in a hyperbolic space this last classification is not wholly satisfactory, for it is at once too inclusive and too ill-defined. It is too inclusive, for we are interested only in real surfaces, and must distinguish between real and imaginary curves of intersection. The Absolute has no real generators, so that we must exclude those cases where the two surfaces cut in one or more generators. It is not sharp enough, for if we search for the shape of our surface, we wish to know whether the curve which it cuts from the Absolute-the absolute curve, let us say-and the corresponding focal developable are real or imaginiary; whether there be real vertices to the common self-conjugate tetrahedron; etc. Furthermore, we care not at all for those surfaces, analyt-

We will show the existence of a duality on $CMC$-1 surfaces in hyperbolic 3-space, and we will show an analogue of the Osserman Inequality in terms of dual surfaces. Moreover, we will show that equality holds (in this analogue) if and only if all the ends of the surface are regular and embedded.

In this note we present a method for constructing constant mean curvature on surfaces in hyperbolic 3-space in terms of holomorphic data first introduced in Bianchi's Lezioni di Geometria Differenziale of 1927, therefore predating by many years the modern approaches due to Bryant, Small and others. Besides its obvious historical interest, this note aims to complement Bianchi's analysis by deriving explicit formulae for CMC-1 surfaces and comparing the various approaches encountered in the literature.

We review recent results on classifying complete constant mean 1 (CMC 1) surfaces in hyperbolic 3-space with low curvature. There are two natural notions of total curvature -- one is the absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual absolute curvature, which is the absolute of the dual CMC 1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce new results as well.

We prove the existence of many constant mean curvature surfaces of revolution with two ends which are immersed or embedded in hyperbolic space. We also study their stability.

We construct a family of Hermitian metrics on the Hopf surface S 3 × S 1 \mathbb {S}^3\times \mathbb {S}^1 , whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of π : S 3 × S 1 → C P 1 \pi :\mathbb {S}^3\times \mathbb {S}^1\to \mathbb {C} P^1 two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces. A similar result is true for S 2 n − 1 × S 1 \mathbb {S}^{2n-1}\times \mathbb {S}^{1} .

In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space $$\mathbb {H}^{n+m}(-1)$$ . That is, we show that a high dimensional complete immersed minimal submanifold M in $$ \mathbb {H}^{n+m}(-1)$$ , is totally geodesic if the $$L^d$$ norm of |A|, for some d, on geodesic balls centered at some point $$p \in M $$ has less than quadratic growth and if either $$\sup _{x \in M} |A|^2$$ is not too large or the $$L^n$$ norm of |A| on M is finite, were, A is the second fundamental form of M.

Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr.7 (2004), 69–100 (electronic)] introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the entropy of the geodesic flow on the associated Riemannian surface. We provide a useful estimate on this function which, in particular, yields a new proof of Bowen’s theorem on the rigidity of the Hausdorff dimension of the limit set of quasi-Fuchsian groups. These follow from new lower bounds on the Hausdorff dimension of the limit set which allow us to give a quantitative version of Bowen’s rigidity theorem. To demonstrate the strength of the techniques, these results are generalized to convex-cocompact surface groups acting on $n$-dimensional $\text{CAT}\,(-1)$ Riemannian manifolds.

We study a complete noncompact minimal submanifold Mn in a sphere Sn+p. We prove there is no nontrivial L2 harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.

Correction to: Stable minimal surfaces and spatial topology in general relativity

Generalized Bernstein Theorem for stable minimal plateau surfaces