Noncompact Spaces Research Articles

Codimension one Ricci soliton subgroups of solvable Iwasawa groups

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces

Let ,G/K, be an irreducible non-compact Hermitian symmetric space and let ,D, be a ,K-invariant domain in ,G/K. In this paper we characterize several classes of ,K-invariant plurisubharmonic functions on ,D, in terms of their restrictions to a slice intersecting all ,K-orbits. As applications we show that ,K-invariant plurisubharmonic functions on ,D, are necessarily continuous and we reproduce the classification of Stein ,K-invariant domains in ,G/K, obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).

Einstein hypersurfaces of Damek-Ricci spaces

Einstein hypersurfaces are “very rare” in rank-one symmetric spaces. Damek-Ricci spaces may be viewed as the closest and the most natural generalisations of noncompact rank-one symmetric spaces. We prove that no Damek-Ricci space admits an Einstein hypersurface.

The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature

In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N N -Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N N -Bakry Émery Ricci curvature. In addition, we show that if M n M^n is a complete, noncompact Riemannian manifold with nonnegative N N -Bakry Émery Ricci curvature where N > n N>n , then H n − 1 ( M , Z ) H_{n-1}(M,\mathbb {Z}) is 0 0 .

An analogue of Pillai's theorem for continued fraction normality and an application to subsequences

We show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to S. S. Pillai in 1940 for base-$b$ expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space. Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality.

A general theory of coexistence and extinction for stochastic ecological communities.

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Homogeneous non-degenerate 3-(\u03b1,\u03b4)-Sasaki manifolds and submersions over quaternionic K\xe4hler spaces

We show that every 3-(alpha ,delta )-Sasaki manifold of dimension 4n + 3 admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature 16n(n+2)alpha delta. In the non-degenerate case we describe all homogeneous 3-(alpha ,delta )-Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If alpha delta > 0, this yields a complete classification of homogeneous 3-(alpha ,delta )-Sasaki manifolds. For alpha delta < 0, we provide a general construction of homogeneous 3-(alpha , delta )-Sasaki manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.

SINGULARITIES OF HOLOMORPHICALLY EXTENDED SPHERICAL FUNCTIONS

1. Motivation and background This is a preliminary account of joint work in progress which serves as an (very) extended abstract for a presentation of one of us ( $E.M$ .O.) in the Awajishima conference on Representation Theory, November 16-19, 2004, Japan. To motivate the questions which we will address we will describe in some detail the beautiful ideas that were initiated by Sarnak [18] and by Bernstein and Reznikov [2], and then further explored by Kr\otz and Stanton [9]. Inspired by Sarnak [18], Bernstein and Reznikov [2] proposed a new method for estimating the coefficients in the expansion of the square of a Maass form on a compact locally symmetric space $Z=\Gamma\backslash X$ (where $X=G/K$ denotes a noncompact Riemannian symmetric space, and $\Gamma\subset G$ is a $co$-compact discrete subgroup of $G$ ) with respect to an orthonormal basis of $L^{2}(Z)$ consisting of Maass forms. The method is based on holomorphic extension of irreducible representations of $G$ to a certain $G$-invariant domain in $X_{\mathbb{C}}:=G_{\mathbb{C}}/If_{\mathbb{C}}$ (we assume that $G\subset G_{\mathbb{C}})$ . In [2] the method was applied in the case of $G=SL_{2}(\mathbb{R})$ . The method was carried further by Kr\otz and Stanton in [9], where the results of [2] were slightly improved for $G=SL_{2}(\mathbb{R})$ , and similar results for other rank 1 Riemannian symmetric spaces $G/K$ were obtained. In addition some higher rank cases were considered in [9]. These considerations gave rise to various interesting issues concerning holomorphic extensions of representations and their matrix coefficients.

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the $G_2$-structure. Then $M/{\langle \iota \rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $\lambda$ on $L$. Much of the analytic difficulty lies in constructing appropriate closed $G_2$-structures with sufficiently small torsion to be able to apply the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi-Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.

Heisenberg spin chain as a worldsheet coordinate for lightcone quantized string

Although the energy spectrum of the Heisenberg spin chain on a circle defined byH=14∑k=1Mσkxσk+1x+σkyσk+1y+Δσkzσk+1z\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ H=\\frac{1}{4}\\sum \\limits_{k=1}^M\\left({\\sigma}_k^x{\\sigma}_{k+1}^x+{\\sigma}_k^y{\\sigma}_{k+1}^y+\\Delta {\\sigma}_k^z{\\sigma}_{k+1}^z\\right) $$\\end{document}is well known for any fixed M, the boundary conditions vary according to whether M ∈ 4ℕ + r, where r = −1, 0, 1, 2, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with M spins breaking into strings with M1< M and M − M1 spins (or vice versa). We organize the energy spectrum and degeneracies of H in the case ∆ = 0 where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit M → ∞ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate x(σ) ≡ x(σ) + R0. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for ∆ ≠ 0 is a change in the compactification radius R0→ R∆. As ∆ → −1 this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795).

Riesz means on symmetric spaces

Let X be a non-compact symmetric space of dimension n. We prove that if f∈Lp(X), 1≤p≤2, then the Riesz means SRz(f) converge to f almost everywhere as R→∞, whenever Rez>(n−12)(2p−1).

Collectively fixed point theorems in noncompact abstract convex spaces with applications

&lt;abstract&gt;&lt;p&gt;In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.&lt;/p&gt;&lt;/abstract&gt;

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.

Hamiltonian and reversible systems with smooth families of invariant tori

For various values of n, d, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any N≥3 and any vector ω∈RN, we present an example of an analytic Hamiltonian system with N degrees of freedom and with an isolated (and even unique) invariant N-torus carrying conditionally periodic motions with frequency vector ω (but this torus is atropic rather than Lagrangian and the symplectic form is not exact). Examples of isolated atropic invariant tori carrying conditionally periodic motions are given in the paper for the first time. The paper can also be used as an introduction to the problem of the isolatedness of invariant tori in Hamiltonian and reversible systems.

Topological Analysis of Fibrations in Multidimensional (C, R) Space

A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.

Minakshisundaram–Pleijel coefficients for non-compact higher rank symmetric spaces

We assign to a non-compact Riemannian symmetric space X a theta function and a zeta function. We compute all of the Minakshisundaram–Pleijel coefficients in a short-time asymptotic expansion of the theta function especially when X is of complex type, which we use to compute the one-loop effective potential—whose relevance for quantum field theory, for example, is briefly commented on. These coefficients, for X of general type, are also shown to be specifically related to residues and special values of the zeta function. The results presented extend previous ones in the literature where the rank of X is assumed to be one.

Surjectivity of convolution operators on noncompact symmetric spaces

Let μ be a K-invariant compactly supported distribution on a noncompact Riemannian symmetric space X=G/K. If the spherical Fourier transform μ˜(λ) is slowly decreasing, it is known that the right convolution operator cμ:f↦f⁎μ maps E(X) onto E(X). In this paper, we prove the converse of this result. We also prove that cμ has a fundamental solution if and only if μ˜(λ) is slowly decreasing.

String baryon in four-dimensional N=2 supersymmetric QCD from the 2D-4D correspondence

We study non-Abelian vortex strings in four-dimensional (4D) $\mathcal{N} = 2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark hypermultiplets. It has been recently shown that these vortices behave as critical superstrings. The spectrum of closed string states in the associated string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N} = 2$ supersymmetric QCD. In particular, the lowest string state appears to be a massless BPS "baryon." Here we show the occurrence of this stringy baryon using a purely field-theoretic method. To this end we study the conformal world-sheet theory on the non-Abelian string -- the so called weighted $\mathcal{N} = (2,2)$ supersymmetric $\mathbb{CP}$ model. Its target space is given by the six-dimensional non-compact Calabi-Yau space $Y_6$, the conifold. We use mirror description of the model to study the BPS kink spectrum and its transformations on curves (walls) of marginal stability. Then we use the 2D-4D correspondence to show that the deformation of the complex structure of the conifold is associated with the emergence of a non-perturbative Higgs branch in 4D theory which opens up at strong coupling. The modulus parameter on this Higgs branch is the vacuum expectation value of the massless BPS "baryon" previously found in string theory.

On generalized compensation functions for factor maps between shift spaces on countable alphabets

We show the existence of generalized compensation functions for a particular type of one-block factor maps [Formula: see text] between countable subshifts [Formula: see text] and [Formula: see text]. For factor maps between compact spaces, continuous compensation functions were studied by Walters in relation to the theory of relative pressure. Applying the thermodynamic formalism for sequences on countable subshifts, we generalize some existing results on factor maps between compact spaces to non-compact spaces. For related questions, we also study the existence of a preimage measure on [Formula: see text] of an invariant measure on [Formula: see text], and their relations.