# Manifolds with analytic corners

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Manifolds with analytic corners

Abstract: Manifolds with boundary and with corners form categories ${\bfMan}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has twonotions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle${}^bTX$. The usual definition of smooth structure uses $TX$, as$f:X\to\mathbb{R}$ is defined to be smooth if $\nabla^kf$ exists as acontinuous section of $\bigotimes^kT^*X$ for all $k\ge 0$. We define 'manifolds with analytic corners', or 'manifolds with a-corners',with a different smooth structure, in which roughly $f:X\to\mathbb{R}$ issmooth if ${}^b\nabla^kf$ exists as a continuous section of$\bigotimes^k({}^bT^*X)$ for all $k\ge 0$. These are different from manifoldswith corners even when $X=[0,\infty)$, for instance$x^\alpha:[0,\infty)\to\mathbb{R}$ is smooth for all real $\alpha\ge 0$ when$[0,\infty)$ has a-corners. Manifolds with a-boundary and with a-corners formcategories ${\bf Man}\subset{\bf Man^{ab}}\subset{\bf Man^{ac}}$, with wellbehaved differential geometry. Partial differential equations on manifolds with boundary may have boundaryconditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumannboundary conditions, or (ii) 'at infinity', prescribing the asymptoticbehaviour of the solution. We argue that manifolds with corners should be usedfor (i), and with a-corners for (ii). We discuss many applications of manifoldswith a-corners in boundary problems of type (ii), and to singular p.d.e.problems involving 'bubbling', 'neck-stretching' and 'gluing'.

Notes:73 pages

Bibliographic Details

Identifiers

Source identifier: 623283 Item Description

Type: General item; Keywords: math.DG math.DG math.AP Tiny URL: pubs:623283

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Autor: Joyce, D - institutionUniversity of Oxford Oxford, MPLS, Mathematical Institute - - - - Bibliographic Details Identifiers Urn: uu