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1,231
result(s) for
"Hyperbolic space"
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Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be
thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the latter one provides a natural
framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on
hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups,
eBook
Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane
The automorphisms of a two-generator free group \\mathsf F_2 acting on the space of orientation-preserving isometric actions of \\mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \\Gamma on \\mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \\kappa _\\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \\kappa _{\\Phi}^{-1}(k).
eBook
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic
metric space. This summarizes and completes a long line of results by many authors, from Patterson’s classic ’76 paper to more recent
results of Hersonsky and Paulin (’02, ’04, ’07). Concrete examples of situations we consider which have not been considered before
include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point
of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater
generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between
Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are
purely geometric: a generalization of a theorem of Bishop and Jones (’97) to Gromov hyperbolic metric spaces, and a proof that the
uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson–Sullivan measure
unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.
eBook
Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane
The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.
Journal Article
Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces
Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the
n
-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).
Journal Article
Geometric pressure for multimodal maps of the interval
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting
of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric
Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized
multimodal maps, that is smooth maps
eBook
Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space
In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of k -th mean curvatures with k=1, n , and positive powers of p -th power sums S_p with p > 0 . We prove that if the initial hypersurface M_0 is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time t > 0 , exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function f of the shifted principal curvatures _i=_i-1 , plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where f is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.
Journal Article
Solitons to mean curvature flow in the hyperbolic 3-space
We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic 3 -space H^3 , providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete translators in H^3 , one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in H^3 and apply it to prove that properly immersed translators to MCF in H^3 are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the z -axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in H^3 which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in H^3 and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in R^3 .
Journal Article
An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds
We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $ P_{\\lambda }:= -\\Delta _{{\\open H}^{N}} - \\lambda $ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of $-\\Delta _{{\\open H}^{N}} $, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator $P_{\\lambda _{1}({\\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\\lambda.$
Journal Article