Narrow Results

This paper deals with global compactness and the multiplicity of positive solutions to problems of the type

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalizes the previous result [S. Guo, Convex hypersurfaces evolving by functions of the mean curvature, preprint (2016), arXiv:1610.08214] for convex hypersurfaces in the Euclidean space by the author to the setting in the hyperbolic space for the same class of flows.

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an $L^2$-almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)].

In this paper we give necessary and sufficient conditions for discreteness of a subgroup of PSL(2,ℂ) generated by a hyperbolic element and an elliptic one of odd order with non-orthogonally intersecting axes. Thus we completely determine two-generator non-elementary Kleinian groups without invariant plane with real traces of the generators and their commutator. We also give a list of all parameters that correspond to such groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold ℍ³/Γ₃₅₃ has real parameters.

In this paper we study group actions on hyperbolic Λ-metric spaces, where Λ is an ordered abelian group. Λ-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicity for such spaces. Only the case of 0-hyperbolic Λ-metric spaces (that is, Λ-trees) was systematically studied, while the theory of general hyperbolic Λ-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from the theory of group actions on hyperbolic (in the usual sense) spaces to the case of Λ-metric spaces for an arbitrary Λ. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them.

We give conditions under which a quasi-isometric map between direct products of hyperbolic spaces splits as a direct product up to bounded distance and permutation of factors. This is a variation on a result due to Kapovich, Kleiner and Leeb.

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic $\mathrm{\Lambda }$-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic $\mathrm{\Lambda }$-metric spaces, where $\mathrm{\Lambda }$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $\mathrm{\Lambda }={\mathbb{Z}}^n$ taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ${\mathbb{Z}}^n$-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy$:$$($AMS-$209)$, Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ${\mathbb{Z}}^n$-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ${\mathbb{Z}}^n$, Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

We study, analytically and by numerical simulations, a mathematical model of closed unknotted polymer chains. The polymers are modeled by what we call "densely packed knots", which are knot diagrams entirely covering a long rectangular lattice, with randomly generated overpasses and underpasses at the nodes of the lattice. We are interested in the probability of the "knottedness" (the "knot complexity") of densily packed knots, which we measure via the degree of their Jones–Kauffman polynomial. In particular, we find the mean complexity of "daughter knots", obtained by cutting off a part of a trivial (i.e. unknotted) "parent" densily packed knot and closig up the "open tails" (the loose ends of the cut strands). We present arguments supporting the conjecture that the knot complexity n^* of a daughter knot of an unknotted parent one grows as where N is the total number of vertices on the lattice. This result gives a strong support for the conjectured "crumpled globule" structure of collapsed unknotted closed polymer chains, in which the polymer forms a system of densely packed folds, mutually separated in a broad range of scales.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices. This verifies affirmatively a conjecture of John Minor.

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with non-positive curvature. We show this result by constructing a non-properly embedded minimal plane in H³. Hence, this gives a counterexample to Calabi–Yau conjecture for embedded minimal surfaces in negative curvature case.

In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δ_𝔹u - λu = |u|^p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.

This paper deals with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in ${ℝ}^N.$ Combining this compactness embedding with the Mountain Pass Theorem, a result of the existence of positive solution is established.

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. On the contrary, there are not so many characterizations of the hyperbolic space, the spacelike sphere in the Minkowski space. By means of a purely synthetic technique, we get a rigidity result for the sphere in 𝔼ⁿ⁺¹ without any curvature conditions, nor completeness or compactness, as well as a dual result for the n-dimensional hyperbolic space in 𝕃ⁿ⁺¹.

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H³. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H³ to S² with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.

We show that in dimensions > 1 the cohomology groups of the Higson compactification of the hyperbolic space ℍⁿ with respect to the C₀ coarse structure are trivial.

Also we prove that the cohomology groups of the Higson compactification of ℍⁿ for the bounded coarse structure are trivial in all even dimensions.

In this paper, we obtain strong convergence results for asymptotically demicontractive and asymptotically hemicontractive mappings in hyperbolic spaces. We present our results in hyperbolic spaces. This class of spaces contains both linear and nonlinear spaces like CAT(0) spaces, $ℝ$-trees, Banach spaces and Hilbert spaces. Thus our results are not only novel but also much more general.

We determine minimal hypersurfaces of the hyperbolic space ℍ⁴(−1) with identically zero Gauss-Kronecker curvature. Such a hypersurface is the image of a subbundle spanned by a timelike vector field of the normal bundle of a totally geodesic surface of the de Sitter space under the normal exponential map. We also give some examples.