Space Hn Research Articles

Conformal solitons for the mean curvature flow in hyperbolic space

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space Hn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{n+1}$$\\end{document}. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field -∂0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\partial _0$$\\end{document}. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of ∂∞Hn+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _\\infty \\mathbb {H}^{n+1}$$\\end{document}. We conclude by proving rigidity results for bowl and grim-reaper cylinders.

Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces

We introduce a type of minimal surface in the pseudo-hyperbolic space Hn,n (with n even) or Hn+1,n−1 (with n odd) coming from cyclic SO0(n,n+1)-Higg bundles. By showing that these surfaces are saddle-type critical points for the area functional, and hence infinitesimally rigid, we get a new proof, for SO0(n,n+1), of Labourie's theorem that the holonomy map restricts to an immersion on the cyclic locus of Hitchin base, and extend it to Collier's components. This implies Labourie's former conjecture in the case of the exceptional group G2′, for which we also show that these minimal surfaces are J-holomorphic curves of a particular type in the almost complex H4,2.

The Christoffel problem in the hyperbolic plane

The horospherical p-Minkowski problem in hyperbolic space Hn+1 was proposed by Li-Xu in [14]. In particular, when n=1,p=−1, it is called the Christoffel problem in the hyperbolic plane. The corresponding equation is given byφ(φθθ−φθ22φ+φ−φ−12)=f(θ),θ∈[0,2π). In this paper, we obtain two existence results of solutions to the above equation.

Equilibria and energy minimizers for an interaction model on the hyperbolic space

We study an intrinsic model for collective behaviour on the hyperbolic space Hn. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of Hn. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a technique used for Rn to establish the existence of compactly supported global minimizers. Numerical simulations are presented, suggesting that some of the equilibria studied here are global attractors. The key tool in our investigations is a family of isometries of Hn that we have developed for this purpose.

Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces and related Riemannian manifolds

We establish sharpened forms of the Hardy type identities and inequalities which are substantial improvements of Hardy inequalities for the operators −ΔH−(N−1)24 on the hyperbolic spaces HN. More precisely, we study the refined Hardy-Poincaré-Sobolev type inequalities in the spirit of Brezis-Vázquez [17] and Brezis-Marcus [15] on hyperbolic spaces, spherically symmetric Riemannian manifolds and more general Riemannian manifolds. Spherically symmetric manifolds are also of significant importance in physics and general relativity. Our approaches are to first establish the Hardy-Poincaré-Sobolev type identities using the notion of Bessel pairs on hyperbolic spaces and related Riemannian manifolds. Using a Hardy identity on upper half space, we also establish a sharp Sobolev inequality on a novel example of noncomplete and non-extendable Riemannian manifold with positive Ricci curvature. In particular, in dimension 3, our example shows that the completeness assumption of the manifolds in the rigidity result of Ledoux [47] cannot be removed.

On the pointwise converse of Fatou’s theorem for Euclidean and real hyperbolic spaces

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space ℝ n+1+ . We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of certain nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn. We shall also prove a generalization of a result regarding large time behavior of a solution of the heat equation proved in [17]. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn.

Fourth Order Schrödinger Equation with Mixed Dispersion on Certain Cartan-Hadamard Manifolds

This paper is devoted to the study of the following fourth order Schrödinger equation with mixed dispersion on MN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^N$$\\end{document}, an N-dimensional Cartan-Hadamard manifold. Namely we consider 4NLSi∂tψ=-ΔM2ψ+βΔMψ+λ|ψ|2σψinR×M,ψ(0,·)=ψ0∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} i\\partial _t\\psi = -\\Delta _M^2\\psi +\\beta \\Delta _M \\psi +\\lambda |\\psi |^{2\\sigma }\\psi \\quad \ ext {in }\\mathbb {R}\ imes M,\\\\ \\psi (0,\\cdot )=\\psi _0\\in X, \\end{array}\\right. } \\end{aligned}$$\\end{document}where β≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\ge 0$$\\end{document}, λ={-1,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =\\{-1,1\\}$$\\end{document}, 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document}, ΔM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _M$$\\end{document} is the Laplace-Beltrami operator on M and X=L2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (M)$$\\end{document} or X=H2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (M)$$\\end{document}. At first, we focus on the case where M is the hyperbolic space HN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^N$$\\end{document}. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to (4NLS) as well as scattering for small initial data provided that N≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 4$$\\end{document} and 0<σ<4/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/N$$\\end{document} if X=L2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L^2 (\\mathbb {H}^N)$$\\end{document} or 0<σ<4/(N-4)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\sigma < 4/(N-4)_+$$\\end{document} if X=H2(HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=H^2 (\\mathbb {H}^N)$$\\end{document}. Next, we obtained weighted Strichartz estimates for radial solutions to (4NLS) on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.

Faber-Krahn inequalities for the Robin Laplacian on bounded domain in Riemannian manifolds

In this paper, we obtain the Faber-Krahn inequality for the first eigenvalue of the Robin Laplacian on bounded domain in Riemannian manifolds whose Ricci curvature satisfies Ricg≥(n−1). The Faber-Krahn inequality also holds for the Robin Laplacian on bounded domain in hyperbolic space Hn.

Higher order Brezis-Nirenberg problem on hyperbolic spaces: Existence, nonexistence and symmetry of solutions

The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators Pk on bounded domains in the hyperbolic space Hn and as well as on the entire hyperbolic space Hn. Among techniques different from the higher order Brezis-Nirenberg problem in Euclidean spaces, one of our main novelties in the study of existence and symmetry is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Maz'ya inequalities and careful study of delicate properties of Green's functions of Pk−λ on hyperbolic spaces which are of independent interests in dealing with such problems. Such Green's functions allow us to obtain the integral representations of solutions and thus avoid using the maximum principle to establish the symmetry of solutions to the higher order Brezis-Nirenberg problem by developing a moving plane method in integral form on hyperbolic spaces in the spirit of the work of [14].

A class of curvature flows expanded by support function and curvature function in the Euclidean space and hyperbolic space

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space Rn+1 with speed uαf−β, where u is the support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For α⩽0<β⩽1−α, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt [16] and Urbas [40] can be recovered by putting α=0 and β=1 in our first result. If the initial hypersurface is convex, this is our previous work [11]. If α⩽0<β<1−α and the ambient space is hyperbolic space Hn+1, we prove that the flow ∂X∂t=(uαf−β−ηu)ν has a longtime existence and smooth convergence to a coordinate slice. The flow in Hn+1 is equivalent (up to an isomorphism) to a re-parametrization of the original flow in Rn+1 case. Finally, we find a family of monotone quantities along the flows in Rn+1. As applications, we give a new proof of a family of inequalities involving the weighted integral of kth elementary symmetric function for k-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in [20].

Global existence and time decay of the non-cutoff Boltzmann equation with hard potential

This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces HN(Rx,v6) with N≥2. But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000).

Algebraic structure and basics of analysis of n-dimensional quaternionic space

In this study, we focused on n-dimensional quaternionic space Hn. To create the module structure, first part is devoted to define a metric depending on the product order relation of Rn. The set of Hn has been rewritten with a different representation of n-vectors. Using this notation, formulations corresponding to the basic operations in Hn are obtained. By adhering these representations, module structure of Hn over the set of real ordered n-tuples is given. Afterwards, we gave limit, continuity and the derivative basics of quaternion valued functions of a real variable.

Infinite norm of the derivative of the solution operator of Euler equations

Through a simple and elegant argument, we prove that the norm of the derivative of the solution operator of Euler equations posed in the Sobolev space Hn, along any base solution that is in Hn but not in Hn+1, is infinite. We also review the counterpart of this result for Navier–Stokes equations at a high Reynolds number from the perspective of fully developed turbulence. Finally, we present a few examples and numerical simulations to show a more complete picture of the so-called rough dependence upon initial data.

Rigidity and stability estimates for minimal submanifolds in the hyperbolic space

In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold Mn in the hyperbolic space Hn+m in order to show that Mn is totally geodesic. We also obtain sharp upper bound estimates for the first eigenvalue of the super stability operator in the case of M is a surface in H4.

A gap theorem for complete submanifolds with parallel mean curvature in the hyperbolic space

Let M be an n(≥7)-dimensional complete submanifold with parallel mean curvature in the hyperbolic space Hn+p, whose mean curvature satisfies H2−1≤0. Denote by A˚ and BR(q) the trace free second fundamental form of M and the geodesic ball of radius R centered at q∈M, respectively. We prove that if lim supR→∞∫BR(q)|A˚|2dMR2=0, and if (∫M|A˚|ndM)2/n+2n(n−2)3n(n−1)H(∫M|A˚|n/2dM)2/n≤C(n), then M is congruent to an n-dimensional hyperbolic space or the Euclidean space Rn. Here C(n) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n=5,6.

The sharp Sobolev type inequalities in the Lorentz–Sobolev spaces in the hyperbolic spaces

Let W1Lp,q(Hn), 1≤q,p<∞ denote the Lorentz–Sobolev spaces of order one in the hyperbolic spaces Hn. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in W1Lp,q(Hn) with 1≤q≤p which generalizes the result in [41] to the setting of Lorentz–Sobolev spaces. Second, we prove several sharp Poincaré–Sobolev type inequalities in W1Lp,q(Hn) with 1≤q≤p<n which generalize the results in [45] to the setting of Lorentz–Sobolev spaces. Finally, we provide the improved Moser–Trudinger type inequalities in W1Ln,q(Hn) in the critical case p=n with 1≤q≤n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Pólya–Szegö type principle in W1Lp,q(Hn) with 1≤q≤p which maybe is of independent interest.

On the nonlinear Schrödinger equation on the Poincaré ball model

In this paper, we prove the existence of a sequence of nonnegative (weak) solutions for the following problem −ΔHu+u=λα(σ)f(u)inBNu∈H1,2(BN),where ΔH denotes the Laplace–Beltrami operator on the Poincaré ball model BN (with N≥3) of the hyperbolic space HN, α∈L1(BN)∩L∞(BN) is a nonnegative and not identically zero radially symmetric potential, f is a suitable continuous function, and λ is a positive real parameter. The analysis developed in this paper combines a compactness embedding result due to Skrzypczak and Tintarev [(2013), Theorem 1.3 and Proposition 3.1], some group-theoretical arguments on the Poincaré ball model BN, and variational methods for smooth functionals defined on the Sobolev space H1,2(BN) associated to the homogeneous Hadamard manifold BN.

New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

Spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space

We investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space Ln+1, for n≥2, spanning a given boundary datum lying on the hyperbolic space ℍn.

Second order Sobolev type inequalities in the hyperbolic spaces

In this paper, we establish several Poincaré–Sobolev type inequalities concerning to the Laplace–Beltrami operator Δg in the hyperbolic space Hn with n≥5. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or the sharp Sobolev inequality in Hn. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for Δg in Hn. As a consequence, we obtain the Poincaré–Sobolev inequality for the second order GJMS operator P2 in Hn. In dimension 4, we obtain an improvement of the sharp Adams inequality and an Adams inequality with exact growth for radial functions.