Harmonic maps of manifolds with boundary pdf file

On the heat equation for harmonic maps from noncompact manifolds. Boundary behavior of harmonic maps on nonsmooth domains and complete negatively curved manifolds by patricio aviles, h choi and mario micallef download pdf 2 mb. Get a printable copy pdf file of the complete article 546k, or click on a page image below to browse page by page. Pdf a note on boundary regularity of subelliptic harmonic maps. For points in the interior or for manifolds without boundary, the. Some remarks on energy inequalities for harmonic maps with. Harmonic mappings into manifolds with boundary numdam.

The theory of the energy functional and its harmonic. Harmonic maps of manifolds with boundary springerlink. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds, the boundary and the. Boundary behavior of harmonic maps on nonsmooth domains and. We find some new sufficient conditions for the uniform convergence of the heat flow, and hence the existence of harmonic maps.

Harmonic mapping to generate harmonic coordinates in regions with boundary. We describe the problem of finding a harmonic map between noncompact. Harmonic maps between riemannian manifolds were first established by james eells and joseph h. Cohomology of harmonic forms on riemannian manifolds. As an application, we prove a di eomorphism property for such harmonic maps in two dimensions. Au of these spaces are canonically homeomorphic, and we will often sup. Introduction let m and n be two riemannian manifolds of dimension m and n. In this article, we study harmonic maps between two complete noncompact manifolds m and n by a heat flow method. Let m be a compact, connected, oriented smooth riemannian nmanifold with nonempty boundary. Journal of functional analysis 99, 293331 1991 boundary behavior of harmonic maps on nonsmooth domains and complete negatively curved manifolds patrick avil department of mathematics, university of illinois at urbanachampaign, urhana, illinois 61801 h. Let m be a compact, connected, oriented, smooth riemannian ndimensional. Introduction the main result of this article is the following. Some remarks on energy inequalities for harmonic maps with potential volker branding in this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear poisson equation can be generalized to harmonic maps with potential between complete riemannian manifolds. By contrast, when m is connected and has nonempty boundary, it is possible for a pform to be harmonic without being both closed and coclosed.

More information on harmonic maps can be found in the following articles and books. Harmonic maps from noncompact riemannian manifolds with. A note on boundary regularity of subelliptic harmonic maps. Choi department of mathematics, university of iowa, iowa city, iowa 52242 and mario micallef department of mathematics, university of. In this note we describe some examples of minimizing harmonic maps between riemannian manifolds with singularities at a free boundary and we discuss the regularity results for minimizing harmonic maps at a free boundary which we have obtained jointly with frank duzaar. A smooth mapping from m to n is called harmonic if it is an extreme value for the energy functional ef 1 l i dfl 2. Browse other questions tagged smooth manifolds manifolds withboundary or ask your own question. Harmonic maps between rotationally symmetric manifolds.

Harmonic maps from noncompact riemannian manifolds with nonnegative ricci curvature outside a compact set volume 124 issue 6 youde wang. A harmonic map will be a critical point of this energy as discussed later. Harmonic riemannian maps on locally conformal kaehler manifolds. Cohomology of harmonic forms on riemannian manifolds with. Buy harmonic maps of manifolds with boundary lecture notes in mathematics. Let rm be an open set, nn a riemannian manifold, x a collection of vector fields on, and f a smooth map from into nn. Pdf file 1055 kb djvu file 239 kb article info and citation. A note on boundary regularity of subelliptic harmonic maps zhou, zhenrong, kodai mathematical journal, 2005 harmonic measure and polynomial julia sets binder, i. Let m be a compact, connected, oriented smooth riemannian n manifold with nonempty boundary. Boundary harmonische abbildung manifold manifolds mannigfaltigkeit randwertproblem equation function. Developments of harmonic maps, wave maps and yangmills. Nevertheless, we have the following boundary regularity theorem. Let m be a smooth, compact riemannian manifold with smooth boundary. Riemannian metrics harmonic maps from manifolds of l.

The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the holder space c k. M,n implies that du x is a linear map from txm to tuxn, i. This functional e will be defined precisely belowone way of understanding it is to imagine that m is made of rubber and n made of marble their shapes given by their respective metrics, and that the map. While trying to understand some regularity results, i thought about the following naive approach for establishing regularity of weakly harmonic maps between riemannian manifolds. Please read our short guide how to send a book to kindle. Harmonic mappings between riemannian manifolds by anand. Harmonic maps into hyperbolic 3 manifolds 609 denotes the space of measured geodesic laminations on m s, o, where o denotes a hyperbolic metric on s. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition. We call f a subelliptic harmonic map if it is a critical point of the energy. Full text full text is available as a scanned copy of the original print version. Developments of harmonic maps, wave maps and yangmills fields into biharmonic maps, biwave maps and biyangmills fields yuanjen chiang auth. Mn between riemannian manifolds m and n is called harmonic if it is a critical point of the dirichlet energy functional. Boundary value problems for energy minimizing harmonic maps.

Harmonic maps of manifolds with boundary, lecture notes 471, springer, 1975. Harmonic maps of manifolds with boundary computer file. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Harmonic maps on locally conformal kaehler manifolds let m,jand n,j be almost complex manifolds. We prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic. Gradient estimates and blowup analysis for stationary harmonic maps. Since solutions to laplaces equation are generically referred to as harmonic functions, we therefore call these coordinates harmonic coordinates, and the deformations they generate harmonic deformations. Qdm is the space of quadratic differentials on af holomorphic with respect to the conformai structure induced by o. Cohomology of harmonic forms on riemannian manifolds with boundary sylvain cappell, dennis deturck, herman gluck, and edward y. The dirichlet problem at infinity is to construct a proper harmonic map with boundary values this paper concerns existence. Harmonic functions for data reconstruction on 3d manifolds. Miller to julius shaneson on the occasion of his 60th birthday 1. Is this approach for establishing regularity of harmonic maps. In section 4 we studied with further details the harmonic maps constructed in theorem 3.

For subelliptic harmonic maps from a carnot group into a riemannian manifold without boundary, we prove that they are smooth near any e\epsilon regular point see definition 1. This integral makes sense if m and n are riemannian manifolds, m is compact, and f is continuously differentiable. Let m and n be two riemannian manifolds of dimension m and n respectively. Harmonic maps from noncompact riemannian manifolds with non. Hermitian harmonic maps from complete hermitian manifolds to. A wiener criterion for w,q harmonic maps into convex balls was established by paulik p by very different methods. As the generalizations of harmonic maps, we now recall the concepts of biharmonic maps and fharmonic maps.