Noncompact Spaces Research Articles

Mean-field theory based on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators

In this paper, we give an expression for canonical transformation group with Grassmann variables, basing on the Jacobi hsp ≔ semidirect sum hN⋊sp(2N,R)C algebra of boson operators. We assume a mean-field Hamiltonian (MFH) linear in the Jacobi generators. We diagonalize the boson MFH. We show a new aspect of eigenvalues of the MFH. An excitation energy arisen from additional self-consistent field parameters has never been seen in the traditional boson mean-field theory. We derive this excitation energy. We extend the Killing potential in the Sp(2N,R)CU(N) coset space to the one in the Sp(2N+2,R)CU(N+1) coset space and make clear the geometrical structure of Kähler manifold, a non-compact symmetric space Sp(2N+2,R)CU(N+1). The Jacobi hsp transformation group is embedded into an Sp(2N+2,R)C group and an Sp(2N+2,R)CU(N+1) coset variable is introduced. Under such mathematical manipulations, extended bosonization of Sp(2N+2,R)C Lie operators, vacuum function and differential forms for extended boson are presented by using integral representation of boson state on the Sp(2N+2,R)CU(N+1) coset variables.

Compatibility of Covariant Quantum Channels with Emphasis on Weyl Symmetry

Compatibility conditions of quantum channels featuring symmetry through covariance are studied. Compatibility here means the possibility of obtaining two or more channels through partial trace out of a broadcasting channel. We see that covariance conditions can be used to simplify compatibility conditions as the broadcasting channel can be assumed to be covariant in a particular way. A particular emphasis is on Weyl covariance and in determining compatibility conditions for Weyl-covariant channels. The concrete examples studied include the case of a non-compact continuous phase space and the case of a finite phase space.

Levi-umbilical real hypersurfaces in Hermitian symmetric spaces

We give a classification of Levi-umbilical real hypersurfaces, whose Levi form is proportional to the induced metric, in the complex two plane Grassmannian G2(Cm+2), the complex quadric Qn, or their noncompact dual spaces.

Entropies of Random Transformations on a Non-compact Space

In this paper, we present several possible natural extensions of metric and topological entropies for random transformations on a non-compact space. In particular, Brin–Katok local entropy formula and Katok $$\delta $$ entropy formula will be reformulated in the context of non-compact setting. Furthermore, a corresponding variational principle for non-compact systems is given.

Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

Abstract Let ℒ {\mathcal{L}} be the Laplace operator on ℝ d {\mathbb{R}^{d}} , d ≥ 3 {d\geq 3} , or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation ℒ ⁢ u - φ ⁢ ( ⋅ , u ) = 0 {\mathcal{L}u-\varphi(\,\cdot\,,u)=0} we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

Remainders of products, topological groups and Cp-spaces

Let X=∏i∈IXi be a product of non-compact spaces. We show that if |I|>ω, then the remainder Y=bX∖X is pseudocompact, for any compactification bX of X. In fact, this theorem follows from a more general result about spaces with an ω-directed lattice of d-open mappings. Under the additional assumption that the space X has countable cellularity, we prove that the remainder Y is C-embedded in bX and that βY=bX. We apply these results to the remainders of topological groups and spaces of continuous functions with the pointwise convergence topology. For example, we prove that if X is an uncountable space and G is a non-compact topological group, then every remainder of Cp(X,G) is pseudocompact provided Cp(X,G) is dense in GX.

A coarse relative-partitioned index theorem

It seems that the index theory for non-compact spaces has found its ultimate formulation in the realm of coarse spaces and K-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this paper we formulate a combination of these two theorems and establish a partitioned-relative index theorem.

On the global sup-norm of GL(3) cusp forms

Let φ be a spherical Hecke–Maas cusp form on the non-compact space PGL3(ℤ)PGL3(ℝ). We establish various pointwise upper bounds for φ in terms of its Laplace eigenvalue λφ. These imply, for φ arithmetically normalized and tempered at the archimedean place, the bound $$||\phi |{|_\infty } \ll_\varepsilon \lambda _\phi ^{39/40 + \varepsilon }$$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the GL3 Jacquet–Whittaker function.

Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space.

On the Entropy Points and Shadowing in Uniform Spaces

We introduce and study the topological concepts of entropy points, expansivity and shadowing property for dynamical systems on noncompact nonmetrizable spaces, which generalize the relevant concepts for metric spaces. We also obtain various properties on uniform entropy on noncompact nonmetrizable spaces. The main result is a theorem which yields a relation between topological shadowing property and positive uniform entropy.

Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space

In this paper, let (Mn,g,dμ) be n-dimensional noncompact metric measure space which satisfies Poincaré inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation△p,fv+cvσ=0, where c≤0,m>n≥1,1<p<m−1+(m−1)(m+3)2,σ≤p−1 are real constants.

Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution

We prove under mild conditions that the Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson processes and diffusion processes with soft killing.

Stochastic approximation on noncompact measure spaces and application to measure-valued Pólya processes

Our main result is to prove almost-sure convergence of a stochastic-approximation algorithm defined on the space of measures on a non-compact space. Our motivation is to apply this result to measure-valued P\'olya processes (MVPPs, also known as infinitely-many P\'olya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochastic-approximation methods to measure-valued Markov processes with a non-compact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs, this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding weights to the different colors of the infinitely-many-color urn. We also exhibit a link between MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the non-balanced case. Finally, we show how our result can be applied to designing stochastic-approximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on non-compact spaces.

The moduli spaces of equivariant minimal surfaces in {mathbb {RH}}^3 and mathbb {RH}^4 via Higgs bundles

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space for the non-compact symmetric space mathbb {RH}^n and show how SO_0(n,1)-Higgs bundles can be used to parametrise this space, making clear how the classical invariants (induced metric and second fundamental form) figure in this picture. We use this parametrisation to provide details of the moduli spaces for mathbb {RH}^3 and mathbb {RH}^4, and relate their structure to the structure of the corresponding Higgs bundle moduli spaces.

Remarks on Metric Entropy of Random $${\mathbb {Z}}^2$$ Z 2 Actions on a Non-compact Space

In this paper, we establish Brin–Katok local entropy and Katok $$\delta $$ entropy formula of random $${\mathbb {Z}}^2$$ transformations on a non-compact space.

Improbability of Wandering Orbits Passing Through a Sequence of Poincaré Surfaces of Decreasing Size

Given a volume preserving dynamical system with non-compact phase space, one is sometimes interested in special subsets of its wandering set. One example from celestial mechanics is the set of initial values leading to collision. Another one is the set of initial values of semi-orbits, whose asymptotic velocity does not exist as a limit. We introduce techniques that can be helpful in showing that these sets are of measure zero: by defining a sequence of hypersurfaces, that are eventually hit by each of those semi-orbits and whose total surface area decreases to zero.

The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

Let M be an irreducible Hermitian symmetric space of compact type, and let $$\omega $$ be its Kahler form. For a triplet $$(p_1,p_2,p_3)$$ of points in M we study conditions under which a geodesic triangle $$\mathcal T(p_1,p_2,p_3)$$ with vertices $$p_1,p_2,p_3$$ can be unambiguously defined. We consider the integral $$A(p_1,p_2,p_3)=\int _\Sigma \omega $$ , where $$\Sigma $$ is a surface filling the triangle $$\mathcal T(p_1,p_2,p_3)$$ and discuss the dependence of $$A(p_1,p_2,p_3)$$ on the surface $$\Sigma $$ . Under mild conditions on the three points, we prove an explicit formula for $$A(p_1,p_2,p_3)$$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.

The adapted hyper-Kähler structure on the crown domain

Let $$\Xi $$ be the crown domain associated with a non-compact irreducible Hermitian symmetric space G / K. We give an explicit description of the unique G-invariant adapted hyper-Kahler structure on $$\Xi $$ , i.e., compatible with the adapted complex structure $$J_\mathrm{ad}$$ and with the G-invariant Kahler structure of G / K. We also compute invariant potentials of the involved Kahler metrics and the associated moment maps.

Rigidity for convex-cocompact actions on rank-one symmetric spaces

When a discrete group admits a convex-cocompact action on a non-compact rank-one symmetric space, there is a natural lower bound for the Hausdorff dimension of the limit set, given by the Ahlfors regular conformal dimension of the boundary of the group. We show that equality is achieved precisely when the group stabilizes an isometric copy of some non-compact rank-one symmetric space on which it acts with compact quotient. This generalizes a theorem of Bonk-Kleiner, who proved it in the case of real hyperbolic space. To prove our main theorem, we study tangents of Lipschitz differentiability spaces that are embedded in a Carnot group. We show that almost all tangents are isometric to Carnot subgroups, at least when they are rectifiably connected. This extends a theorem of Cheeger, who proved it for PI spaces that are embedded in Euclidean space.