Class Of Non-compact Manifolds Research Articles

Boutet de Monvel operators on Lie manifolds with boundary

We introduce and study a general pseudodifferential calculus for boundary value problems on a class of non-compact manifolds with boundary (so-called Lie manifolds with boundary). This is accomplished by constructing a suitable generalization of the Boutet de Monvel calculus for boundary value problems. The data consists of a compact manifold with corners M that is endowed with a Lie structure of vector fields 2V, a so-called Lie manifold. The manifold M is split into two equal parts X+ and X− which intersect in an embedded hypersurface Y⊂X±. Our goal is to describe a transmission Boutet de Monvel calculus for boundary value problems compatible with the structure of Lie manifolds. Starting with the example of b-vector fields, we show that there are two groupoids integrating the Lie structures on M and on Y, respectively. These two groupoids form a bibundle (or a groupoid correspondence) and, under some mild assumptions, these groupoids are Morita equivalent. With the help of the bibundle structure and canonically defined manifolds with corners, which are blow-ups in particular cases, we define a class of Boutet de Monvel type operators. We then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem. Finally, we consider appropriate Fredholm conditions and construct the parametrices for elliptic operators in the calculus.

On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour, forλ→+∞, of the counting functionNP(λ)of certain positive self-adjoint operatorsPwith double order(m,μ),m,μ>0, m≠μ, defined on a manifold with endsM. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined onℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae forNP(λ)and show how their behaviour depends on the ratiom/μand the dimension ofM.

Wodzicki residue for operators on manifolds with cylindrical ends

We define the Wodzicki Residue TR(A) for A in a space of operators with double order (m_1,m_2). Such operators are globally defined initially on R^n and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function. Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator belonging the mentioned class in the case m_1=m_2.

Complex Powers and Non-compact Manifolds

We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

Local models for intersecting brane worlds

We describe the construction of configurations of D6-branes wrapped on compact 3-cycles intersecting at points in non-compact Calabi-Yau threefolds. Such constructions provide local models of intersecting brane worlds, and describe sectors of four-dimensional gauge theories with chiral fermions. We present several classes of non-compact manifolds with compact 3-cycles intersecting at points, and discuss the rules required for model building with wrapped D6-branes. The rules to build 3-cycles are simple, and allow easy computation of chiral spectra, RR tadpoles and the amount of preserved supersymmetry. We present several explicit examples of these constructions, some of which have Standard Model like gauge group and three quark-lepton generations. In some cases, mirror symmetry relates the models to other constructions used in phenomenological D-brane model building, like D-branes at singularities. Some simple N=1 supersymmetric configurations may lead to relatively tractable G_2 manifolds upon lift to M-theory, which would be non-compact but nevertheless yield four-dimensional chiral gauge field theories.

Asymptotics of Dirichlet Spectrum on Some Class of Noncompact Domains

We investigate the asymptotic behaviour of the counting function of Dirichlet eigenvalues on some class of noncompact manifolds. We prove that in cases when the volume or the perimeter (the volume of the boundary) of the manifold is infinite, some additional (nonclassical) terms appear in the precise asymptotics. The coefficients at the classical terms in those are regularized in some special way volume (resp. perimeter) of the manifold.

Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

Consider the scaling invariant, sharp log entropy (functional) introduced by Weissler \cite{W:1} on noncompact manifolds with nonnegative Ricci curvature. It can also be regarded as a sharpened version of Perelman's W entropy \cite{P:1} in the stationary case. We prove that it has a minimizer if and only if the manifold is isometric to $\R^n$. Using this result, it is proven that a class of noncompact manifolds with nonnegative Ricci curvature is isometric to $\R^n$. Comparing with the well known flatness results in \cite{An:1}, \cite{Ba:1} and \cite{BKN:1} on asymptotically flat manifolds and asymptotically locally Euclidean (ALE) manifolds, their decay or integral condition on the curvature tensor is replaced by the condition that the metric converges to the Euclidean one in $C^1$ sense at infinity. No second order condition on the metric is needed.