Groupoids and pseudodifferential calculus on manifolds with corners

We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of C∗-algebras of groupoids, we also obtain new proofs for the study of b-calculus.     

Pseudodifferential calculus on manifolds with corners and groupoids

We build a longitudinally smooth, differentiable groupoid associated to any manifold with corners. The pseudodifferential calculus on this groupoid coincides with the pseudodifferential calculus of Melrose (also called -calculus). We also define an algebra of rapidly decreasing functions on this groupoid; it contains the kernels of the smoothing operators of the (small) -calculus.

     

Transversality on infinite dimensional manifolds with corners

In this paper we study transversality on infinite dimensional manifolds with corners. We deal with transversality, boundary-transversality and weak-boundary-transversality, characterize these three notions by means of the infinitesimal transversality, construct submanifolds as inverse image of submanifolds by transversal maps and prove parametrized theorems about the density of weak-boundary-transversality and boundary-transversality, which generalize the corresponding Abraham and Quinn theorems.     

On cobordism of manifolds with corners

This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.

     

Symbolic calculus for boundary value problems on manifolds with edges

Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbolic structure is responsible for ellipticity and for the nature of parametrices within an algebra of “edge-degenerate” pseudo-differential operators. The edge symbolic component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operator-valued Mellin symbols. We establish a calculus in a framework of “twisted homogeneity” that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.     

Applications of proximal calculus to fixed point theory on Riemannian manifolds

We prove a general form of a fixed point theorem for mappings from a Riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum (when a Lie group structure is given on the manifold) and composition. In order to prove our main result we develop a theory of proximal calculus in the setting of Riemannian manifolds.     

Hilbert manifolds with corners of finite codimension and the theory of optimal control

This survey presents a version of Palais-Smale theory for Hilbert manifolds that is convenient for investigation of optimal control problems associated with smooth control systems of constant rank. Results obtained with A. A. Agrachev concerning the finite-dimensional case — the analog of Morse theory for (finite-dimensional) manifolds with corners are given. Simple applications of the theory are discussed. A necessary condition for global controllability of systems of constant rank is obtained, as well as a dual result on the multiplicity of solutions for the corresponding optimization problems.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 96–171, 1990.     

A calculus for ideal triangulations of three‐manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three‐manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,α), where M is a three‐manifold and α is a collection of properly embedded arcs. We also show that certain well‐understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,α). Our proof does not assume the Matveev–Piergallini calculus for ideal triangulations, and actually easily implies this calculus. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hinged and supported plates with corners

We consider the Kirchhoff–Love model for the supported plate, that is, the fourth-order differential equation Δ² u?=?f with appropriate boundary conditions. Due to the expectation that a downwardly directed force f will imply that the plate, which is supported at its boundary, touches that support everywhere, one commonly identifies those boundary conditions with the ones for the so-called hinged plate: u?=?0?=?Δu ? (1 ? σ ) κ u _n . Structural engineers however are usually aware that rectangular roofs tend to bend upwards near the corners, and this would mean that u?=?0 is not appropriate. We will confirm this behavior and show the difference of the supported and the hinged plates in case of domains with corners.     

Singular integral operators on curves with corners

The symbol map of Gohberg and Krupnik [6] for the closed algebra generated by singular integral operators with piecewise continuous coefficients is extended to the case of curves with corners. This algebra includes the operator of the double layer potential on such curves.     

The Boggess-Polking extension theorem for<Emphasis Type="Italic">CR</Emphasis> functions on manifolds with corners

We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ ⁿ withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T _NM in a neighborhood of a pointz _o.We identify in natural manner and assume that for a holomorphic vector fieldL tangent toM and verifying we have that the Levi form takes a value . Then we prove thatCR functions onV extend ∀_ω to a wedgeV ₁ “attached” toV in direction of a vector fieldiV such that |pr(iV(z ₀))−iv ₀| < ε (where pr is the projection pr:T _NX →T _MX | _N ).We then prove that when the Levi cone “relative to Σ”iZ _Σ = convex hull is open inT _MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT _NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ _Σ =T _MX, and is continuous up toN, thenf is in fact holomorphic atz _o.     

Reconstructing curves with sharp corners

In this paper we present a heuristic to reconstruct nonsmooth curves with multiple components. Experiments with several input data reveals the effectiveness of the algorithm in contrast with the other competing algorithms.