Noncompact Spaces Research Articles

Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: A computational approach

Let G be a simple non-compact linear connected Lie group and H⊂G be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non-virtually abelian subgroups Γ⊂G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup Γ⊂G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup, provided that the pair (g,h) is contained in a database created by De Graaf and Marrani.

Topological Weak Specification and Distributional Chaos on Noncompact Spaces

In this paper, we relate the topological definitions of specification property and distributional chaos defined for uniformly continuous self-maps on noncompact, nonmetrizable spaces. We prove that a uniformly continuous surjective self-map acting on a uniformly locally compact Hausdorff uniform space with topologically weak specification property and a pair of distal points is topologically distributionally chaotic of type 1. This extends the result due to Oprocha and Štefánková [2008]. As a consequence, we get that uniformly continuous surjective self-map on a uniformly locally compact totally bounded Hausdorff uniform space with topological shadowing, topological mixing, and a distal pair is topologically distributionally chaotic of type 1.

Infinite Energy Quasi-Periodic Solutions to Nonlinear Schrödinger Equations on ℝ

Abstract We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schrödinger equations on $\mathbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous results [3, 4], and the generalization [32], are quasi-periodic in time, but periodic in space. This paper generalizes J. Bourgain’s [5] semi-algebraic set method to analyze nonlinear PDEs, in the non-compact space quasi-periodic setting on $\mathbb R$.

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

Error estimates for DeepONets: a deep learning framework in infinite dimensions

Abstract DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ordinary differential equation, an elliptic partial differential equation (PDE) with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon $ is argued to suffer from a ‘curse of dimensionality’ (requiring a neural networks of exponential size in $1/\epsilon $), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon $ with neural networks of size that can grow algebraically in $1/\epsilon $).Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

Explicit Proper p-Harmonic Functions on the Riemannian Symmetric Spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n)

In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian symmetric spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n). We also describe how the same can be manufactured on their non-compact symmetric dual spaces.

Ultraviolet sensitivity of Casimir energy

Abstract We quantitatively estimate the effect of ultraviolet (UV) physics on the Casimir energy in a five-dimensional (5D) model on S1/Z2. If the cutoff scale of the 5D theory is not far from the compactification scale, the UV physics may affect the low-energy result. We work in the cutoff regularization scheme by introducing two independent cutoff scales for the spatial momentum in the non-compact space and for the Kaluza–Klein masses. The effects of the UV physics are incorporated as a damping effect of the contributions to the vacuum energy around the cutoff scales. We numerically calculate the Casimir energy and evaluate the deviation from the result obtained in the zeta-function regularization, which does not include information on the UV physics. We find that the result agrees well with the latter for Gaussian-type damping, while it can deviate for kink-type damping.

Ergodic Optimization and Zero Temperature Limits in Negative Curvature

In this paper, we study aspects of the ergodic theory of the geodesic flow on a non-compact negatively curved manifold. In topological dynamics, it is a well-known fact that every continuous potential on a compact metric space has a maximizing measure. Unfortunately, for non-compact spaces this fact is no longer true. For the geodesic flow we provide a criterion that ensures the existence of a maximizing measure for uniformly continuous potentials. We prove that the only obstruction to the existence of a maximizing measure is the full escape of mass phenomenon. To the best of our knowledge, this is the first general result on the existence of maximizing measures for non-compact topological spaces which does not require the potential to be coercive. We study zero temperature limits of equilibrium measures for a suitable family of potentials. We prove some convergence and divergence results for the limiting behavior of such measures. Among some consequences, we obtain that the geodesic flow has the intermediate entropy property and that equilibrium states are dense in the space of invariant probability measures.

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with \operatorname{Ric}_{\infty} \ge 1 . Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry–Ledoux’s Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag’s needle decomposition (also called localization), and is inspired by a recent work of Cavalletti, Maggi and Mondino on compact spaces. Besides the quantitative isoperimetry, a reverse Poincaré inequality for the guiding function that we have as a key step, as well as the way we use it, are of independent interest.

Differentiability of the pressure in non-compact spaces

Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of

Taxonomy of brane gravity localisations

Generating an effective theory of lower-dimensional gravity on a submanifold within an original higher-dimensional theory can be achieved even if the reduction space is non-compact. Localisation of gravity on such a lower-dimensional worldvolume can be interpreted in a number of ways. The first scenario, Type I, requires a mathematically consistent Kaluza-Klein style truncation down to a theory in the lower dimension, in which case solutions purely within that reduced theory exist. However, that situation is not a genuine localisation of gravity because all such solutions have higher-dimensional source extensions according to the Kaluza-Klein ansatz. Also, there is no meaningful notion of Newton’s constant for such Type I constructions.Types II and III admit coupling to genuinely localised sources in the higher-dimensional theory, with corresponding solutions involving full sets of higher-dimensional modes. Type II puts no specific boundary conditions near the worldvolume aside from regularity away from sources. In a case where the wave equation separated in the non-compact space transverse to the worldvolume admits a normalisable zero mode, the Type III scenario requires boundary conditions near the worldvolume that permit the inclusion of that zero mode in mode expansions for gravitational wave fluctuations or potentials. In such a case, an effective theory of lower-dimensional gravity can emerge at sufficiently large worldvolume distance scales.This taxonomy of brane gravity localisations is developed in detail for linearised perturbations about a background incorporating the vacuum solution of Salam-Sezgin theory when embedded into ten-dimensional supergravity with a hyperbolic non-compact transverse space. Interpretations of the Newton constant for the corresponding Type III localisation are then analysed.

The isoperimetric problemviadirect method in noncompact metric measure spaces with lower Ricci bounds

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N) spaces (X, d, ℋN). Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov-Hausdorff limits of the ambient spaceXalong diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of setsEi⊂Xiwith uniformly bounded measure and perimeter, where (Xi,di, ℋN) is an arbitrary sequence of RCD(K,N) spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.

Marcinkiewicz-type multipliers on products of noncompact symmetric spaces

In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.

On the geometry of irreversible metric-measure spaces: Convergence, stability and analytic aspects

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.

AdS2 geometries and non-Abelian T-duality in non-compact spaces

We obtain an AdS2 solution to Type IIA supergravity with 4 Poincaré supersymmetries, via non-Abelian T-duality with respect to a freely acting SL(2,R) isometry group, operating on the AdS3×S3×CY2 solution to Type IIB. That is, non-Abelian T-duality on AdS3. The dual background obtained fits in the class of AdS2×S3×CY2 solutions to massive Type IIA constructed in [1]. We propose and study a quiver quantum mechanics dual to this solution that we interpret as describing the backreaction of the baryon vertex of a D4-D8 brane intersection.

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

We introduce various new type of recurrent sets for finitely generated semigroups on non-compact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

Some Properties of the Topological Entropy of a Family of Dynamical Systems Defined on an Arbitrary Metric Space

We consider a family of dynamical systems defined on a noncompact metric space andcontinuously depending on a parameter varying in some metric space. For any such family, thetopological entropy of the dynamical systems in the family is studied as a function of theparameter from the viewpoint of the Baire classification of functions.

Wave equation on certain noncompact symmetric spaces

In this paper, we prove sharp pointwise kernel estimates and dispersive properties for the linear wave equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

The Alekseevskii conjecture in 9 and 10 dimensions

We show that non-compact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is to show, via a cohomogeneity-one approach, that non-compact homogeneous spaces admitting an ideal isomorphic to sl2(R) admit no homogeneous Einstein metrics of negative Ricci curvature.

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of q-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.