Non-compact Case Research Articles

Divergence theorems and the supersphere

The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge–Hilgert–Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and show how superharmonic functions are related to conserved quantities. An integration over the supersphere was introduced by Coulembier–De Bie–Sommen as a generalisation of the Pizzetti integral. In this context, a mean value theorem for harmonic superfunctions was established. We formulate this integration along the lines of the general theory and give a superior proof of two mean value theorems based on our divergence theorem.

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in Olmos and Reggiani (J. Reine Angew. Math. 664:29–53, 2012) to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.

The web of D-branes at singularities in compact Calabi-Yau manifolds

We present novel continuous supersymmetric transitions which take place among different chiral configurations of D3/D7 branes at singularities in the context of type IIB Calabi-Yau compactifications. We find that distinct local models which admit a consistent global embedding can actually be connected to each other along flat directions by means of transitions of bulk-to-flavour branes. This has interesting interpretations in terms of brane recombination/splitting and brane/anti-brane creation/annihilation. These transitions give rise to a large web of quiver gauge theories parametrised by splitting/recombination modes of bulk branes which are not present in the non-compact case. We illustrate our results in concrete global embeddings of chiral models at a dP_0 singularity.

Stackelberg equilibria via variational inequalities and projections

Existence and location of Stackelberg equilibria is studied for two players by using appropriate variational inequalities and fixed point arguments. Both compact and non-compact strategy sets are considered in Euclidean spaces; in the non-compact case, we apply arguments from the theory of (discrete and continuous) projective dynamical systems. Some examples are also presented.

Extremal of Log Sobolev inequality and W entropy on noncompact manifolds

Let M be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (1.1) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This result has the following consequences. 1. It seems to give the first example of connected, complete manifolds with bounded geometry where a standard Log Sobolev inequality does not have an extremal. 2. It gives a negative answer to the open question on the existence of extremal of Perelmanʼs W entropy in the noncompact case, which was stipulated by Perelman (2002) [22, p. 9, 3.2 Remark]. 3. It helps to prove, in some cases, that noncompact shrinking breathers of Ricci flow are gradient shrinking solitons.

Heegaard genera in congruence towers of hyperbolic 3-manifolds

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $\epsilon>0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-\epsilon}$. We get similar results in the arithmetic non-compact case.

Boundary behaviour of special cohomology classes arising from the Weil representation

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

New results on noncompact harmonic manifolds

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z. I. Szabó [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. In this paper we study, using a notion of rank, the asymptotic geometry and the geodesic flow on simply connected nonflat and noncompact harmonic manifolds denoted by X . In the first part of the paper we show that the following assertions are equivalent. The volume growth is purely exponential, the rank of X is one, the geodesic flow is Anosov with respect to the Sasaki metric, X is Gromov hyperbolic. We also provide a characterization of manifolds of constant negative curvature by their minimal volume growth among those harmonic manifolds with fixed mean curvature of the horospheres. In the second part of the paper we show that the geodesic flow is Anosov if X is a nonflat harmonic manifold with no focal points. In the course of the proof we obtain that certain partially hyperbolic flows on arbitrary Riemannian manifolds without focal points are Anosov, which is of interest beyond harmonic manifolds. Combining the results of this paper with the rigidity theorem’s of [BCG], [BFL], and [FL], we confirm the Lichnerowicz conjecture for all compact harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups.

Impure aspects of supersymmetric Wilson loops

We study a general class of supersymmetric Wilson loops operator in N = 4 super Yang-Mills theory, obtained as orbits of conformal transformations. These loops are the natural generalization of the familiar circular Wilson-Maldacena operator and their supersymmetric properties are encoded into a Killing spinor that is not pure. We present a systematic analysis of their scalar couplings and of the preserved supercharges, modulo the action of the global symmetry group, both in the compact and in the non-compact case. The quantum behavior of their expectation value is also addressed, in the simplest case of the Lissajous contours: explicit computations at weak-coupling, through Feynman diagrams expansion, and at strong-coupling, by means of AdS/CFT correspondence, suggest the possibility of an exact evaluation.

A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define L\'evy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.

EXISTENCE OF EQUILIBRIA IN COMPLETE METRIC SPACES

In this paper, we establish equilibrium version of Ekeland's variational principle without assuming any kind of semicontinuity of the bifunction involved in the formulation of the principle. By using such principle, we derive some existence results for a solution of equilibrium problems with or without compactness assumption on the underlying set. A coercivity condition is introduced to obtain a solution of an equilibrium problem for noncompact case. Our results extend and improve several known results in the literature.

Ricci Flow on Open 4-Manifolds with Positive Isotropic Curvature

In this note we prove the following result: Let X be a complete, connected 4-manifold with uniformly positive isotropic curvature, with bounded geometry, and with no essential incompressible space form. Then X is diffeomorphic to $\mathbb{S}^{4}$ , or $\mathbb{RP}^{4}$ , or $\mathbb{S}^{3}\times\mathbb {S}^{1}$ , or $\mathbb{S}^{3}\widetilde{\times} \mathbb{S}^{1}$ , or a possibly infinite connected sum of them. This extends work of Hamilton and Chen–Zhu to the noncompact case. The proof uses Ricci flow with surgery on complete 4-manifolds, and is inspired by recent work of Bessieres, Besson, and Maillot.

Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes

AbstractIn this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

Time adaptive variational integrators: A space–time geodesic approach

The goal of this paper is to show that the space–time geodesic approach of classical mechanics can be used to generate a time adaptive variational integration scheme. The only assumption we make is that the Lagrangian for the system is in a separable form. The geometric structure which is preserved in the integration scheme is made explicit and the algorithm is illustrated with simulation for a compact case, a non-compact case, a chaotic system which arises as a perturbation of an integrable system and the figure eight solution for a three body problem.

Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound

In this article we are concerned with the strong stabilization of models for the Reissner–Mindlin plate equations with second sound, that is, models that include thermal effects described according to Cattaneo's law of heat conduction instead of Fourier's law in classical thermoelasticity. Two models will be considered which are distinct with respect to the property of compactness or non-compactness of the resolvent of the generator of the underlying semigroup. In accordance with the compactness or non-compactness of the resolvent operator, a different criterion for strong stability is implemented to achieve the strong stabilization of each model. In the compact resolvent case we avail ourselves of a result given by Benchimol [C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), pp. 373–379] and in the non-compact case we resort to a stability criterion Tomilov [Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63–98].

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.

Some Applications of the Hodge-de Rham Decomposition to Ricci Solitons

The aim of this paper is to present a link between the Perelman potential for a compact Ricci soliton Mn and the Hodge-de Rham decomposition theorem, we shall use this result to present an integral formula which enables us to establish conditions under which the Ricci soliton is trivial. Moreover, given a Ricci soliton such that its associated vector field X is a conformal vector field we show that in the compact case X is a Killing vector field, while for the non-compact case, either the soliton is Gaussian or X is a Killing vector field.

Interior capacities of condensers with countably many plates in locally compact spaces

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a countable, locally finite collection of arbitrary sets with the sign $+1$ or $-1$ prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in $\mathbb R^n$, $n\geqslant2$, and closed condensers of finitely many plates). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed, unbounded condensers of infinitely many plates). For all positive definite kernels satisfying Fuglede's condition of consistency between the strong and vague ($={}$weak$*$) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in $\mathbb R^n$, which is important in applications.

Interior Capacities of Condensers in Locally Compact Spaces

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a finite collection of arbitrary sets with sign + 1 or − 1 prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in \(\mathbb R^n\) and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying Fuglede’s condition of consistency between the strong and vague (= weak*) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We describe those solutions and the corresponding equilibrium constants, analyze their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in \(\mathbb R^n\), which is important in applications.

Rigidity of shrinking Ricci solitons

We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.