Noncompact Spaces Research Articles

Non-Compact Complex Projective Space as a Phase Space

First of all we give a simple example of Kahler manifolds (pseudo)Euclidean complex spaces as well as compact and non-compact complex projective spaces. Moreover we describe The Klein model of Lobachevsky space and mapping a conformal mechanics to itin one dimensional case first. Than we extend Klein model for the N-dimensional. We got interesting results that isometry generators of the phase space, namely Killing potentials are appear to be the constants of motion of the system.

Schwinger effect in compact space: A real time calculation

We compute the discharging rate of a uniform electric field due to Schwinger pair production in ($1+1$)-dimensional scalar electrodynamics with a compact dimension of radius $R$. Our calculation is performed in real time, using the in-in formalism. For large compactification radii, $R\ensuremath{\rightarrow}\ensuremath{\infty}$, we recover the standard noncompact space result. However, other ranges of values of $R$ and of the mass $m$ of the charged scalar give rise to a richer set of behaviors. For $R\ensuremath{\gtrsim}\mathcal{O}(1/m)$ with $m$ large enough, the electric field oscillates in time, whereas for $R\ensuremath{\rightarrow}0$ it decreases in steps. We discuss the origin of these results.

Indecomposable continua as Higson coronae

In this paper, we consider spaces whose Higson coronae are indecomposable continua. We show that for a non-compact proper metric space X which is coarsely geodesic and has coarse bounded geometry, the Higson corona of X is an indecomposable continuum if and only if X is coarsely equivalent to the space of natural numbers. Then we give characterizations of finitely generated groups that have one or two ends by decomposability/indecomposability of the components of their Higson coronae.

No-scale and scale-separated flux vacua from IIA on G2 orientifolds

We discuss flux compactifications of IIA string theory on G2 holonomy spaces with O2/O6-planes to three dimensions and find two classes of solutions: (1) No-scale Minkowski vacua from NSNS 3-form fluxes and RR 4-form fluxes. (2) By adding Romans mass we find AdS_3 vacua for which the AdS scale can be decoupled completely from the KK scale while the solution is at tunable weak coupling and large volume. For the AdS_3 vacuum we only have a proper 3D description (i.e. smeared orientifold description) of the solution like the 4D analogue of IIA moduli stabilization from fluxes and O6-planes. This 3D description reveals that moduli with non-compact moduli spaces can be stabilized at the classical level. For both types of vacua we can have supersymmetry or not.

Weak Shadowing for Actions of Some Finitely Generated Groups on Non-compact Spaces and Related Measures

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter’s stability theorem to group actions on locally compact metric spaces. We show that action $\varphi :\mathbb {F}_{2}\times M\to M$ of free group F2 = on compact manifold M with dim(M) ≥ 2 has shadowing property if and only if it has weak shadowing property. This implies that if $\varphi :\mathbb {F}_{2}\times M\to M$ is topologically stable, then it has shadowing property. Finally, we introduce a measure $\mu \in {\mathscr{M}}(X)$ that is compatible with the weak shadowing property for the continuous action φ : G × X → X, denoted by $\mu \in {\mathscr{M}}(X, \varphi )$. We show that if φ can be approximated by μ-transitive actions for some $\mu \in {\mathscr{M}}(X, \varphi )$, then Ω(φ) = X. We give an example to show that it may be ${\mathscr{M}}(X, \varphi )\neq \emptyset $ while φ does not have weak shadowing property but we show that if there is a strictly measure $\mu \in {\mathscr{M}}(X, \varphi )$, then continuous action φ of G on compact metric space X has weak shadowing property. Also, we show that if G is a finitely generated abelian group and ${\mathscr{M}}(X, \varphi )\neq \emptyset $, then closure of ${\mathscr{M}}(X, \varphi )$ has an φ-invariant measure.

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush–Kuhn–Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash’s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces.

Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation $${\Delta _{p,f}}u + c{u^\sigma } = 0$$ on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ (1 < p ≤ 2, σ ≤ p - 1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality.

Mass gaps and braneworlds*

Remembering the foundational contributions of Peter Freund to supergravity, and especially to the problems of dimensional compactification, reduction is considered with a non-compact space transverse to the lower dimensional theory. The known problem of a continuum of Kaluza–Klein states is avoided here by the occurrence of a mass gap between a single normalizable zero-eigenvalue transverse wavefunction and the edge of the transverse state continuum. This style of reduction does not yield a formally consistent truncation to the lower dimensional theory, so developing the lower-dimensional effective theory requires integrating out the Kaluza–Klein states lying above the mass gap.

η-Einstein Sasakian immersions in non-compact Sasakian space forms

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $$ {\mathbb{B}}^N\times {\mathbb{R}}$$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the η-Einstein case.

Heterotic instantons for monad and extension bundles

We consider non-perturbative superpotentials from world-sheet instantons wrapped on holomorphic genus zero curves in heterotic string theory. These superpotential contributions feature prominently in moduli stabilization and large field axion inflation, which makes their presence or absence, as well as their functional dependence on moduli, an important issue. We develop geometric methods to compute the instanton superpotentials for heterotic string theory with monad and extension bundles. Using our methods, we find a variety of examples with a non-vanishing superpotential. In view of standard vanishing theorems, we speculate that these results are likely to be attributed to the non-compactness of the instanton moduli space. We test this proposal, for the case of monad bundles, by considering gauged linear sigma models where compactness of the instanton moduli space can be explicitly checked. In all such cases, we find that the geometric results are consistent with the vanishing theorems. Surprisingly, linearly dependent Pfaffians even arise for cases with a non-compact instanton moduli space. This suggests some gauged linear sigma models with a non-compact instanton moduli space may still have a vanishing instanton superpotential.

Area minimizing discs in locally non-compact metric spaces

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.

Embeddings from noncompact symmetric spaces to their compact duals

Every compact symmetric space $M$ admits a dual noncompact symmetric space $\check{M}$. When $M$ is a generalized Grassmannian, we can view $\check{M}$ as a open submanifold of it consisting of space-like subspaces \cite{HL}. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric R-spaces. We will compare these embeddings and describe their images using cut loci.

Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces

The aim of this work is to study homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. First, we deduce a formula for Ricci tensor of a homogeneous pseudo-Riemannian manifold with compact isotropy subgroup. Based on this formula, we establish a one-to-one correspondence between homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces and homogeneous Riemannian Einstein metrics on compact homogeneous spaces. As an application, we prove that every noncompact connected simple Lie group except SL(2) admits at least two nonproportional left invariant pseudo-Riemannian Einstein metrics. Furthermore, we study left invariant pseudo-Riemannian Einstein metrics on solvable Lie groups. By showing that if a nilpotent Lie group admits a left invariant Riemannian Ricci soliton, then it admits a left invariant pseudo-Riemannian Ricci soliton as well, we construct a left invariant pseudo-Riemannian Einstein metric on any given Riemannian standard Einstein solvmanifold.

Topological unified [formula omitted]-entropy

In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified (r,s)-entropy. The properties of topological entropy and Shannon entropy are similar.In this paper, we extend the concept of unified (r,s)-entropy to the topological dynamical system by using Bowen’s definition of separated and spanning sets. We call this notion topological Unified (r,s)-entropy. Then we find the value of topological unified (r,s)-entropy when X is a noncompact metric space and get the value of the topological unified (r,s)-entropy for X∗ when X∗ is one point-compactification of X.

English

We show that n-dimensional (n ⩾ 2) complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure n-space (i.e. the Euclidean metric n-space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.

Uniqueness of Nonnegative Solutions to Elliptic Differential Inequalities on Finsler Manifolds

We consider a class of elliptic differential inequalities involving Finsler p-Laplacian and a positive potential function on forward geodesically complete noncompact Finsler measure spaces with finite reversibility. Under various volume growth conditions concerning geodesic balls with a given center and the potential function, we prove that the only nonnegative weak solution of the differential inequalities is identically zero.

One-loop determinants for black holes in 4d gauged supergravity

We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d mathcal{N} = 2 gauged supergravity, initiated in [1]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space ℍ2× Σg with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.

Metric distortion in the geometric Schottky problem

The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus \(J(\mathcal{M}_g)\) in the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties as an algebraic subvariety. By viewing \(\mathcal{A}_g\) as a noncompact metric space coming from its structure as a locally symmetric space and \(J(\mathcal{M}_g)\) as a metric subspace, we compare the subspace metric d and the induced length metric l on \(J(\mathcal{M}_g)\). Consequently, we clarify the nature of the metric distortion of the subspace \(J(\mathcal{M}_g)\) and hence settle a problem posed by Farb (2006) on the metric distortion of \(J(\mathcal{M}_g)\) inside \(\mathcal{A}_g\) in a certain sense (see Theorem 1.5 and Corollary 1.6).

Existence of solutions of set-valued equilibrium problems in topological vector spaces with applications

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

Upper bounds for the spectral function on homogeneous spaces via volume growth

We use spectral embeddings to give upper bounds on the spectral function of the Laplace–Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also improve Li’s bound itself. Our bounds translate to explicit upper bounds on the heat kernel for both compact and noncompact homogeneous spaces.