CONFIGURATION CATEGORIES AND HOMOTOPY AUTOMORPHISMS

AbstractLet M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).

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The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Chinese Annals of Mathematics Series B ◽

10.1007/s11401-015-0924-6 ◽

2015 ◽

Vol 36 (6) ◽

pp. 991-1000 ◽

Cited By ~ 1

Author(s):

Jingchen Hu ◽

Yiqian Shi ◽

Bin Xu

Keyword(s):

Compact Manifold ◽

Gradient Estimate ◽

Manifold With Boundary

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A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

International Journal of Mathematics ◽

10.1142/s0129167x18500696 ◽

2018 ◽

Vol 29 (10) ◽

pp. 1850069

Author(s):

Hong Zhang

Keyword(s):

Asymptotic Behavior ◽

Boundary Value Problem ◽

Mean Curvature ◽

Compact Manifold ◽

Boundary Value ◽

Fundamental Solutions ◽

Curvature Equation ◽

Mean Curvature Equation ◽

Manifold With Boundary ◽

Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

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Homology stability for symmetric diffeomorphism and mapping class groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000638 ◽

2015 ◽

Vol 160 (1) ◽

pp. 121-139 ◽

Cited By ~ 1

Author(s):

ULRIKE TILLMANN

Keyword(s):

Compact Manifold ◽

Smooth Manifold ◽

Mapping Class Groups ◽

Mapping Class ◽

Classifying Spaces ◽

Class Groups ◽

Connected Sum ◽

Smooth Compact Manifold ◽

Embedded Disks ◽

Homology Stability

AbstractFor any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

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Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

International Mathematics Research Notices ◽

10.1093/imrn/rnz157 ◽

2019 ◽

Author(s):

Michael Levitin ◽

Alexander Strohmaier

Keyword(s):

Finite Element Method ◽

Numerical Experiments ◽

Compact Manifold ◽

Good Accuracy ◽

Scattering Matrix ◽

Direct Computation ◽

Simple Method ◽

Manifold With Boundary ◽

Arithmetic Surfaces ◽

Genus One

Abstract In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM. All the videos accompanying this paper are available with its online version, or externally either at michaellevitin.net/hyperbolic.html or as a dedicated YouTubeplaylist.

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Bistable vector fields are axiom A

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013915 ◽

1995 ◽

Vol 51 (1) ◽

pp. 83-86

Author(s):

Mike Hurley

Keyword(s):

Vector Field ◽

Compact Manifold ◽

Vector Fields ◽

Axiom A ◽

Smooth Compact Manifold ◽

Structurally Stable

Recently L. Wen showed that if a C1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.

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Some embeddings of projective spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034083 ◽

1959 ◽

Vol 55 (4) ◽

pp. 294-298 ◽

Cited By ~ 10

Author(s):

I. M. James

Keyword(s):

Euclidean Space ◽

Compact Manifold ◽

Projective Spaces ◽

Dimensional Euclidean Space ◽

Smooth Compact Manifold

1. Introduction. We say that a smooth compact manifold is embedded in q–space when there is given a regular one-one map of the manifold into q–dimensional Euclidean space. Whitney (5) has shown that embeddings always exist when q is not less than twice the dimension of the manifold.

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RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULAS

Journal of Topology and Analysis ◽

10.1142/s1793525309000151 ◽

2009 ◽

Vol 01 (03) ◽

pp. 207-250 ◽

Cited By ~ 2

Author(s):

PIERRE ALBIN ◽

RICHARD MELROSE

Keyword(s):

Compact Manifold ◽

Pseudodifferential Operators ◽

Chern Character ◽

Explicit Formulas ◽

Manifold With Boundary ◽

Index Bundle ◽

Normal Operators ◽

K Theory ◽

Index Formulas ◽

Character Mapping

For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have "geometric K-theory", namely the "transmission algebra" introduced by Boutet de Monvel [5], the "zero algebra" introduced by Mazzeo in [9, 10] and the "scattering algebra" from [16], we give explicit formulas for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fiber operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values.

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Duality by reproducing kernels

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203206037 ◽

2003 ◽

Vol 2003 (6) ◽

pp. 327-395 ◽

Cited By ~ 6

Author(s):

A. Shlapunov ◽

N. Tarkhanov

Keyword(s):

Differential Operator ◽

Neumann Problem ◽

Compact Manifold ◽

Reproducing Kernels ◽

Elliptic Differential Operator ◽

Regularity Theorem ◽

Neumann Problems ◽

Smooth Compact Manifold ◽

Convexity Properties ◽

The Neumann Problem

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.

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