Non-compact Case Research Articles

Symplectic Stability on Manifolds with Cylindrical Ends

The notion of Eliashberg–Gromov convex ends provides a natural restricted setting for the study of analogs of Moser’s symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak–Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and completeness assumptions on the symplectic forms at infinity, we show that symplectic stability holds under a natural growth condition on the path of symplectic forms. The result can be straightforwardly applied as we show through explicit examples.

A fixed point theorem on noncompact manifolds

We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant $K$-theory and $K$-homology, and a non-localised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.

The annoying null boundaries

We consider a class of globally hyperbolic space-times with “expanding singularities”. Under suitable assumptions we show that no C0-extensions across a compact boundary exist, while the boundary must be null wherever differentiable (which is almost everywhere) in the non-compact case.

Linearization instability for generic gravity in AdS spacetime

In general relativity, perturbation theory about a background solution fails if the background spacetime has a Killing symmetry and a compact spacelike Cauchy surface. This failure, dubbed as {\it linearization instability}, shows itself as non-integrability of the perturbative infinitesimal deformation to a finite deformation of the background. Namely, the linearized field equations have spurious solutions which cannot be obtained from the linearization of exact solutions. In practice, one can show the failure of the linear perturbation theory by showing that a certain quadratic (integral) constraint on the linearized solutions is not satisfied. For non-compact Cauchy surfaces, the situation is different and for example, Minkowski space having a non-compact Cauchy surface, is linearization stable. Here we study, the linearization instability in generic metric theories of gravity where Einstein's theory is modified with additional curvature terms. We show that, unlike the case of general relativity, for modified theories even in the non-compact Cauchy surface cases, there are some theories which show linearization instability about their anti-de Sitter backgrounds. Recent $D$ dimensional critical and three dimensional chiral gravity theories are two such examples. This observation sheds light on the paradoxical behavior of vanishing conserved charges (mass, angular momenta) for non-vacuum solutions, such as black holes, in these theories.

Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

Abstract It is shown that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. In the non-compact case, it is proved that a seven-dimensional everywhere non-umbilical qc-hypersurface embedded in a hyper-Kähler manifold is qc-conformal to a qc-Einstein structure which is locally qc-equivalent to the 3-Sasakian sphere, the quaternionic Heisenberg group or the hyperboloid.

Ricci curvature and orientability

In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between $L^2$-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and the Gromov-Hausdorff convergence, which is a generalization of a recent work by Matveev-Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.

Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case

We study the topology of the Liouville foliation of the Goryachev integrable case in the rigid body dynamics which is a one-parameter family of completely integrable Hamiltonian systems with two degrees of freedom. For this problem P. E. Ryabov has found a real separation of variables with the aid of which he studied the phase topology of the Goryachev systems for positive values of the parameter. We solve the similar problem for negative values of the parameter. This case is of special interest because all the leaves of the Liouville foliation and the surfaces of constant energy turn out to be non-compact. The results are presented in the form of Fomenko invariants for all regular energy levels.

Hardy-type results on the average of the lattice point error term over long intervals

Suppose D D is a suitably admissible compact subset of R k \mathbb {R}^k having a smooth boundary with possible zones of zero curvature. Let R ( T , θ , x ) = N ( T , θ , x ) − T k v o l ( D ) R(T,\theta ,x)= N(T,\theta ,x) - T^{k}\mathrm {vol}(D) , where N ( T , θ , x ) N(T,\theta ,x) is the number of integral lattice points contained in an x x -translation of T θ ( D ) T\theta (D) , with T > 0 T >0 a dilation parameter and θ ∈ S O ( k ) \theta \in SO(k) . Then R ( T , θ , x ) R(T,\theta ,x) can be regarded as a function with parameter T T on the space E ∗ + ( k ) E_{*}^{+}(k) , where E ∗ + ( k ) E_{*}^{+}(k) is the quotient of the direct Euclidean group by the subgroup of integral translations and E ∗ + ( k ) E_{*}^{+}(k) has a normalized invariant measure which is the product of normalized measures on S O ( k ) SO(k) and the k k -torus. We derive an integral estimate, valid for almost all ( θ , x ) ∈ E ∗ + ( k ) (\theta ,x) \in E_{*}^{+}(k) , one consequence of which in two dimensions is that for almost all ( θ , x ) ∈ E ∗ + ( 2 ) (\theta ,x) \in E_{*}^{+}(2) , a counterpart of the Hardy circle estimate ( 1 / T ) ∫ 1 T | R ( t , θ , x ) d t | ≪ T 1 4 + ϵ (1/T)\int _{1}^{T} |R(t,\theta ,x)\,dt| \ll T^{\frac {1}{4} +\epsilon } is valid with an improved estimate. We conclude with an account of hyperbolic versions for which, drawing on previous work of Hill and Parnovski, we give counterparts in all dimensions for both the compact and non-compact finite volume cases.

Deformations of some open submanifolds of Calabi–Yau manifolds

Abstract By using L 2 analytic methods we prove some results about holomorphic deformations of some open submanifolds of Calabi–Yau manifolds, which generalize some classical results from compact to noncompact cases.

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.

Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces

We prove the sharp local L^1-L^infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p-L^infty estimate for p>1. It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.

NONCOMPACT GROUPS OF HERMITIAN SYMMETRIC TYPE AND FACTORIZATION

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group G0 of Hermitian symmetric type. For compact groups root subgroup factorization is related to Bott–Samelson desingularization, and many striking applications have been discovered by Lu ([5]). In this paper, in the noncompact Hermitian symmetric case, we obtain parallel characterizations of the Birkhoff components of G0 and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.

Analyzing plasmid segregation: Existence and stability of the eigensolution in a non-compact case

We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.

Effective equidistribution of twisted horocycle flows and horocycle maps

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of Venkatesh, recently already improved by Tanis and Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by Shah and Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cusp forms which are comparable to the best known bounds of Good in the holomorphic case, and of Bernstein and Reznikov in the Maass (non-holomorphic) case. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.

G 2-Monopoles with Singularities (Examples)

$G_2$-monopoles are solutions to gauge theoretical equations on $G_2$-manifolds. If the $G_2$-manifolds under consideration are compact, then any irreducible $G_2$-monopole must have singularities. It is then important to understand which kind of singularities $G_2$-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.

An Urysohn-type theorem under a dynamical constraint

We address the following question raised by M. Entov and L. Polterovich: given a continuous map $f:X\to X$ of a metric space $X$, closed subsets $A,B\subset X$, and an integer $n\geq 1$, when is it possible to find a continuous function $\theta:X\to\mathbb{R}$ such that \[ \theta f-\theta\leq 1, \quad \theta|A\leq 0, \quad\text{and}\quad \theta|B> n\,? \] To keep things as simple as possible, we solve the problem when $A$ is compact. The non-compact case will be treated in a later work.

Onofri inequalities and rigidity results

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

SU(2)/SL(2) knot invariants and Kontsevich–Soibelman monodromies

We review the Reshetikhin–Turaev approach for constructing noncompact knot invariants involving Rmatrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich–Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials.

Proper Modifications of Generalized p-Kähler Manifolds

In this paper, we consider a proper modification $$f : \tilde{M} \rightarrow M$$ between complex manifolds, and study when a generalized p-Kahler property goes back from M to $$\tilde{M}$$ . When f is the blow-up at a point, every generalized p-Kahler property is conserved, while when f is the blow-up along a submanifold, the same is true for $$p=1$$ . For $$p=n-1$$ , we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kahler manifolds.

Forced linear oscillators and the dynamics of Euclidean group extensions

We study the generic dynamical behaviour of skew-product extensions generated by cocycles arising from equations of forced linear oscillators of special form. This work extends our earlier work on cocycles into compact Lie groups arising from differential equations of special form, (cf. [21]), to the case of non-compact fiber groups of Euclidean type. The earlier techniques do not work in the non-compact case. In the non-compact case one of the main obstacle is the lack of `recurrence'. Thus, our approach to studying Euclidean group extensions is : (i) first, to use a `twisted version' of the so called `conjugation approximation method' and then (ii) to use `geometric-control theoretic methods' developed in our earlier work (cf. [20] and [21]). Even then, our arguments only work for base flows that admit a global Poincae section, (e.g. for the irrational rotation flows on tori and for certain nil flows). We apply these results to study generic spectral behaviour of the forced quantum harmonic oscillator with time dependent stationary force restricted to satisfy given constraints.