Non-compact Case Research Articles

Llarull type theorems on complete manifolds with positive scalar curvature

In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold ( M n , g ) (M^{n},g) with scalar curvature R g ≥ 6 R_{g}\geq 6 admits a non-zero degree and 1 1 -Lipschitz map to ( S 3 × T n − 3 , g S 3 + g T n − 3 ) (\mathbb {S}^{3}\times \mathbb {T}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {T}^{n-3}}) , for 4 ≤ n ≤ 7 4\leq n\leq 7 , then ( M n , g ) (M^{n},g) is locally isometric to S 3 × T n − 3 \mathbb {S}^{3}\times \mathbb {T}^{n-3} . Similar results are established for non-compact cases as ( S 3 × R n − 3 , g S 3 + g R n − 3 ) (\mathbb {S}^{3}\times \mathbb {R}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {R}^{n-3}}) being model spaces. We observe that the results differ significantly when n = 4 n=4 compared to n ≥ 5 n\geq 5 . Our results imply that the ϵ \epsilon -gap length extremality of the standard S 3 \mathbb {S}^3 is stable under the Riemannian product with R m \mathbb {R}^m , 1 ≤ m ≤ 4 1\leq m\leq 4 (see D 3 D_{3} . Question in Gromov’s paper [Foundations of mathematics and physics one century after Hilbert, Spring, Cham, 2018, p. 153]).

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

Existence results and equilibrium stability conditions to fuzzy [formula omitted]-player generalized multiobjective games with application to economic equilibrium models

The purpose of this article is to study existence conditions and generic stability of solution sets to the fuzzy generalized multiobjective games in infinite dimensional spaces. First, we revisit a new class of the fuzzy generalized multiobjective games. Second, we introduce the concept of the extended Ci-quasi-concavity and establish existence conditions for fuzzy generalized multiobjective games using Browder type fixed point theorem in the noncompact cases. Third, we establish some sufficient conditions for generic solution stability conditions to such n-player multiobjective games under uncertainty. Finally, as a real-world application, we consider the special case of economic equilibrium models. Existence conditions and generic stability of solution sets for these real models are also obtained. The results obtained in this paper are new and different from the main results given by some authors in the literature.

On the noncollapsedness of positively curved Type I ancient Ricci flows

AbstractIn this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three‐dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.

Light-Scattering Properties for Aggregates of Atmospheric Ice Crystals within the Physical Optics Approximation

This paper presents the light-scattering matrices of atmospheric-aggregated hexagonal ice particles that appear in cirrus clouds. The aggregates consist of the same particles with different spatial orientations and numbers of these particles. Two types of particle shapes were studied: (1) hexagonal columns; (2) hexagonal plates. For both shapes, we studied compact and non-compact cases of particle arrangement in aggregates. As a result, four sets of aggregates were made: (1) compact columns; (2) non-compact columns; (3) compact plates; and (4) non-compact plates. Each set consists of eight aggregates with a different number of particles from two to nine. For practical reasons, the bullet-rosette and the aggregate of hexagonal columns with different sizes were also calculated. The light scattering matrices were calculated for the case of arbitrary spatial orientation within the geometrical optics approximation for sets of compact and non-compact aggregates and within the physical optics approximation for two additional aggregates. It was found that the light-scattering matrix elements for aggregates depend on the arrangement of particles they consist of.

Optimal control of generalized multiobjective games with application to traffic networks modeling

AbstractThe purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder‐type fixed point theorem in the noncompact case and the ‐quasi‐concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended ‐convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real‐world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.

Counting sheaves on Calabi–Yau 4-folds, I

Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for SO(2n,C)-bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided.

Variants of Two-Measure Input-to-State Stability

For attracting closed sets that are not compact we consider different formulations of input-to-state stability properties and related Lyapunov criteria. In the compact case, there exists a well-established hierarchy of such properties and some formulations have been quickly discarded as equivalent to input-to-state stability. In this paper we show for the noncompact case, several new phenomena appear. In particular, input-to-state stability (ISS) does not imply integral input-to-state stability, and ISS is not equivalent to integral-input-to-integral-state stability. The criteria are formulated in terms of measurement functions, which allows a uniform presentation of a number of related results.

On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

Positive scalar curvature on foliations: The noncompact case

Let (M,gTM) be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and F an integrable subbundle of TM. Let kF be the leafwise scalar curvature associated to gF=gTM|F. We show that if either TM or F is spin, then inf(kF)≤0. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Benameur-Heitsch et al. to the noncompact situation.

On generalized quasi-Einstein manifolds

A rigidity result for a class of compact generalized quasi-Einstein manifolds with constant scalar curvature is obtained. Moreover, under some geometric assumptions, the rigidity for the non-compact case is also proved. Considering non constant scalar curvature, we characterize the generalized quasi-Einstein manifolds which is conformal to the Euclidean space and we show that there exist two classes of complete manifolds, which are obtained by considering potential functions and conformal factors either to be radial or invariant under the action of an (n-1) dimensional translation group. Explicit examples are given.

A note on convergence of noncompact nonsingular solutions of the Ricci flow

We extend some convergence results on nonsingular compact Ricci flows in the papers by Hamilton [Comm. Anal. Geom. 7 (1999), pp. 695–729], Sesum [Math. Res. Lett. 12 (2005), pp. 623–632] and Fang, Zhang, and Zhang [J. Geom. Anal. 20 (2010), pp. 592–608] to certain infinite volume noncompact cases which are “partially” nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.

Full Laplace spectrum of distance spheres in symmetric spaces of rank one

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a single unified formula for all cases. As an application, we find all resonant radii for distance spheres in the compact case, i.e., radii where there is bifurcation of embedded constant mean curvature spheres, and show that distance spheres are stable and locally rigid in the noncompact case.

Winding number of a Brownian particle on a ring under stochastic resetting

We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach a steady state, which is in contrast with the non-compact case of a Brownian particle under resetting on the real line. The mean total number of turns and the variance of the net number of turns grow linearly with time, with a proportionality constant equal to the inverse of the mean first-completion time of a turn.

Urysohn-type theorem under a dynamical constraint: Non-compact case

We address the following question raised by M. Entov and L. Polterovich: Suppose $ V $ a vector field on the manifold $ M $ that generates a complete flow $ \varphi_t:M\to M $. If $ A,B\subset M $ are closed, and $ T>0 $, when is it possible to find a smooth function $ \theta:X\to \mathbb{R} $ such that$ V\theta\leq 1\text{, }\theta|A\leq 0\text{, and }\theta|B>T, $where $ V\theta $ is the derivative of $ \theta $ in the direction of $ V $. We solved this problem with $ A $ compact in [5]. In this work, we generalize to the case where $ A $ is not compact.

Eventually expansive semiflows

<p style='text-indent:20px;'>A semiflow is <i>eventually expansive</i> if there is a prefixed radius up to which two orbits eventually coincide. We prove the following result: Every injective eventually expansive semiflow of a compact metric space consists of finitely many closed orbits and, in the noncompact case, the semiflow cannot have global attractors. The suspension of a continuous map is eventually expansive only if the map is positively expansive. The suspension of every positively expansive map is an eventually expansive semiflow. The topological entropy of an eventually expansive semiflow is bounded from below by the growth rate of the periodic orbits. We present some related examples.</p>

Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

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.

Stability of the cut locus and a central limit theorem for Fréchet means of Riemannian manifolds

We obtain a central limit theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin’s Omnibus central limit theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.