Harmonic maps of manifolds with boundary pdf free

In particular, this proves the existence of harmonic maps between such manifolds. We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. A wiener criterion for w,q harmonic maps into convex balls was established by paulik p by very different methods. If m and m are analytic riemann manifolds, then harmonic maps are analytic. Pdf a note on boundary regularity of subelliptic harmonic maps. 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. Furthermore we prove that any continuous map can be free homotopically deformed to a. Mn between riemannian manifolds m and n is called harmonic if it is a critical point of the dirichlet energy functional. 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. Hamilton, harmonic maps of manifolds with boundary, lecture notes in mathematics, vol.

Interior and free boundary regularity for diracharmonic maps. S rm satisfying the mixed boundary conditions for s and this. A critical point of the energy e over ck is a harmonic map with free boundary u. Recently, they have become powerful tools in the study of global properties of riemannian and kahlerian manifolds. Au of these spaces are canonically homeomorphic, and we will often sup. Heat flow and boundary value problem for harmonic maps numdam. Existence of minmax free boundary disks releasing the width of a. 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. Quantitative regularity for p harmonic maps aaron naber. Ii boundary values of bergmanharmonic maps sorin dragomir2 abstract. Manifold with boundary synonyms, manifold with boundary pronunciation, manifold with boundary translation, english dictionary definition of manifold with boundary.

Harmonic mapping to generate harmonic coordinates in regions with boundary. The harmonic map problem with mixed boundary conditions panayotissmyrnelis communicatedbyjamese. Let m be a compact, connected, oriented, smooth riemannian ndimensional. Let us put our results into the context of the av ailable literature. Pdf in this paper, we establish the uniqueness of heat flow of harmonic maps into. Boundary value problems for energy minimizing harmonic maps. Cohomology of harmonic forms on riemannian manifolds with boundary sylvain cappell, dennis deturck, herman gluck, and edward y. We prove partial boundary regularity, namely singu0, a result that was until now only known in the interior of the domain see b. Get a printable copy pdf file of the complete article 546k, or click on a page image below to browse page by page. The theory of the energy functional and its harmonic. Boundary behavior of harmonic maps on nonsmooth domains and.

We give an elementary introduction to cr and pseudohermitian geometry, starting from h. 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. More information on harmonic maps can be found in the following articles and books. 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. A standard reference for this subject is a pair of reports, published in 1978 and 1988 by james eells and luc lemaire. Harmonic maps of surfaces and applications to free boundary problems. 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. 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. Springer nature is making coronavirus research free. The aim is obtaining minkowskitypevolume estimates on the singular set. A harmonic map will be a critical point of this energy as discussed later.

Dirichlet energy of harmonic maps into manifolds nwith nonpositive sectional curvatures. Full text full text is available as a scanned copy of the original print version. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in h10, t. Harmonic maps of manifolds with boundary lecture notes in. I introduction to cr geometry and subelliptic harmonic maps. Jun 25, 2008 boundary regulari of p harmonic maps with free and partially constrained boundary conditions 103 in case g2 b and e r k, the function co can be chosen to satisfy the estimate fb ivcoip dx with nonempty boundary. Harmonic mappings between riemannian manifolds by anand. Let b1 be the unit open disk in r2 and m be a closed riemannian manifold. I introduction1 to cr geometry and subelliptic harmonic maps.

As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact riemannian manifold n without boundary. 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. We also mention earlier works on regularity of harmonic maps at a free boundary, all of which. S1 on a closed, oriented 3manifold, it was shown in 15 that the average euler characteristic r. Buy harmonic maps of manifolds with boundary lecture notes in mathematics. As an application, we solve the dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. Harmonic maps from noncompact riemannian manifolds with nonnegative ricci curvature outside a compact set volume 124 issue 6 youde wang. Miller to julius shaneson on the occasion of his 60th birthday 1. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed dirichlet and free boundaries from. Then a necessary and sufficient condition that there existf a. B1,m whose energy is nonincreasing in time, given initial data u0. Namely, for the level sets u 1f gof a nonconstant harmonic circlevalued map u. Morse theory by perturbation methods with applications to harmonic.

Qdm is the space of quadratic differentials on af holomorphic with respect to the conformai structure induced by o. R2 andrm with the samenumber of sides, we prove the existence and uniqueness of a smooth harmonic map u. The harmonic map problem with mixed boundary conditions. Pdf harmonic map heat flow with rough boundary data. Harmonic maps between rotationally symmetric manifolds. Boundary harmonische abbildung manifold manifolds mannigfaltigkeit randwertproblem equation function. Cohomology of harmonic forms on riemannian manifolds with. In section 4 we studied with further details the harmonic maps constructed in theorem 3. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition.

We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds, the boundary and the. Let and be smooth riemannian manifolds, of the dimension n. Cohomology of harmonic forms on riemannian manifolds. We prove the existence and uniqueness of harmonic maps between rotationally symmetric manifolds that are asymptotically hyperbolic. The heat flow and harmonic maps between complete manifolds. M,n implies that du x is a linear map from txm to tuxn, i. Pdf boundary regularity and the dirichlet problem for. Pdf on uniqueness of heat flow of harmonic maps researchgate. As an application, we prove a di eomorphism property for such harmonic maps in two dimensions. Introduction the main result of this article is the following.

The dirichlet problem at infinity is to construct a proper harmonic map with boundary values this paper concerns existence. Let m, g be a compact riemannian manifold with smooth boundary and n, h be a. Minmax minimal disks with free boundary in riemannian manifolds. In a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds. Partial regularity for stationary harmonic maps at a free. Interior and free boundary regularity for diracharmonic maps, harmonic maps and related pde harmonic maps for maps u. We call f a subelliptic harmonic map if it is a critical point of the energy. Harmonic maps into singular spaces and padic superrigidity. Harmonic maps from noncompact riemannian manifolds with non. Aug 19, 2005 in this case, all the maps in the complex harmm,d are zero, and so hpharmm,d harmpm hpm. A note on boundary regularity of subelliptic harmonic maps. The qualitative behavior at the free boundary for approximate. Harmonic mappings into manifolds with boundary numdam. As the generalizations of harmonic maps, we now recall the concepts of biharmonic maps and fharmonic maps.