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

2015 ◽  
Vol 36 (6) ◽  
pp. 991-1000 ◽  
Author(s):  
Jingchen Hu ◽  
Yiqian Shi ◽  
Bin Xu
Keyword(s):  
Compact Manifold ◽  
Gradient Estimate ◽  
Manifold With Boundary

scholarly journals CONFIGURATION CATEGORIES AND HOMOTOPY AUTOMORPHISMS

2018 ◽  
Vol 62 (1) ◽  
pp. 13-41
Author(s):  
MICHAEL S. WEISS
Keyword(s):  
Compact Manifold ◽  
Geometric Conditions ◽  
Homeomorphism Group ◽  
Manifold With Boundary ◽  
Smooth Compact Manifold

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).

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

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.

scholarly journals Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

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.

RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULAS

2009 ◽  
Vol 01 (03) ◽  
pp. 207-250 ◽  
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.

On the Structure of Semigroups on a Compact Manifold With Boundary

1957 ◽  
Vol 65 (1) ◽  
pp. 117 ◽  
Author(s):  
Paul S. Mostert ◽  
Allen L. Shields
Keyword(s):  
Compact Manifold ◽  
Manifold With Boundary

scholarly journals A Perturbation of the de Rham Complex

2020 ◽  
pp. 519-532
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Euler Characteristic ◽  
Compact Manifold ◽  
Cohomology Theory ◽  
The Other ◽  
De Rham Complex ◽  
Manifold With Boundary ◽  
Other Hand ◽  
Rham Complex ◽  
De Rham ◽  
Natural Way

We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings

scholarly journals Logarithms and sectorial projections for elliptic boundary problems

2008 ◽  
Vol 103 (2) ◽  
pp. 243 ◽  
Author(s):  
Anders Gaarde ◽  
Gerd Grubb
Keyword(s):  
Compact Manifold ◽  
Elliptic Boundary ◽  
Zeta Functions ◽  
Boundary Problems ◽  
Wodzicki Residue ◽  
Manifold With Boundary ◽  
Special Cases ◽  
The Difference ◽  
Definition Of ◽  
Spectral Cut

On a compact manifold with boundary, consider the realization $B$ of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say $\mathsf{R}_{-}$. In the first part of the paper we define and discuss in detail the operator $\log B$; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, $\zeta (B,0)$, and it enters in an important way in studies of composed zeta functions $\zeta (A,B,s)= {\operatorname {Tr}}(AB^{-s})$ (pursued elsewhere). There is a similar definition of the operator $\log_{\theta}B$, when the spectral cut is at a general angle $\theta$. When $B$ has spectral cuts at two angles $\theta <\varphi$, one can define the sectorial projection $\Pi_{\theta,\varphi} (B)$ whose range contains the generalized eigenspaces for eigenvalues with argument in $\left]\theta,\varphi \right[$; this is studied in the last part of the paper. The operator $\Pi_{\theta ,\varphi}(B)$ is shown to be proportional to the difference between $\log_{\theta}B$ and $\log_{\varphi} B$, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general.

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