Nharmonic maps of manifolds with boundary pdf merger

We first establish a bochnerreilly formula for such maps and deduce therefrom some immediate isolation results. Hamilton was able to extend their theory to manifolds with boundary hm. A wiener criterion for w,q harmonic maps into convex balls was established by paulik p by very different methods. This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. Diffusion maps use a parameter, e, which is usually the order of the distance. A smooth structure on such a manifold with boundary is an equivalence class. A little more precisely it is a space together with a way of identifying it locally with a euclidean space which is compatible on overlaps. Qdm is the space of quadratic differentials on af holomorphic with respect to the conformai structure induced by o. On the existence of hermitianharmonic maps from complete hermitian to complete riemannian manifolds. In addition, using a weighted sobolev inequality obtained in, we prove that, under some energy. The solution manual is written by guitjan ridderbos.

The topology of 3manifolds, heegaard distance and the. A note on boundary regularity of subelliptic harmonic maps zhou, zhenrong, kodai mathematical journal, 2005 harmonic measure and polynomial julia sets binder, i. Curvature conditions and statement of results suppose m is a complex manifold with kihler metric ds 2rea p g, dzadzp. More information on harmonic maps can be found in the following articles and books. Introduction to 3manifolds arizona state university. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition. To overcome these difficulties, we will combine some ideas from. Harmonic riemannian maps on locally conformal kaehler manifolds. Feb 16, 2001 we generalize biharmonic maps between riemannian manifolds into the case of the domain being v manifolds. Let us put our results into the context of the av ailable literature. Let m be a compact, connected, oriented, smooth riemannian ndimensional. Diffusion maps 2 have proved to be a powerful tool for analyzing point sets in a highdimensional space for identifying clusters and manifolds, but sometimes they provide a poor parameterization of the manifold near its boundaries, as we will demonstrate. Hermitian harmonic maps from complete hermitian manifolds to.

Harmonic maps between rotationally symmetric manifolds. The heisenberg group denoted by 7id is obtained as cd x m with the group law. Hopefully the results will be useful to study corresponding rigidity of complete hermitian manifolds. Uniqueness theorems for harmonic maps into metric spaces. A standard reference for this subject is a pair of reports, published in 1978 and 1988 by.

In the theory of surfaces, it is well known that the laplacebeltrami operator on compact surfaces. Vmo maps between compact ndimensional oriented manifolds without boundaries. Let b1 be the unit open disk in r2 and m be a closed riemannian manifold. We say that u is constant if its image contains only a single point. One motivation for this generalization is to prove theorem. Most of our arguments will be on b x, so we define for. Harmonic maps into singular spaces and padic superrigidity.

Regularity theory for pharmonic mappings between riemannian. Given a compact smooth manifold m with nonempty boundary and a morse function, a pseudogradient morsesmale vector. Very recently, bethuel b proved partial regularity for 2 harmonic maps with monotonicity for an arbitrary target manifold. Victor goryunov local invariants of maps between 3 manifolds. On the existence of hermitianharmonic maps from complete. Toprepare for the proof of theorem 1, we consider boundary conditions on di. Dlldoc, and the triple d, d, doo is identified topologically with the euclidean triple b, b, sx.

M n be a harmonic map, where n is a cartanhadamard manifold. Heat flow of extrinsic biharmonic maps from a four. Harmonic maps on locally conformal kaehler manifolds let m,jand n,j be almost complex manifolds. Global lightlike manifolds and harmonicity bejan, c. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds, the boundary and the data. We extend their existence and uniqueness results to the case where both domain and target manifolds are complete. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds. Analysis on manifolds lecture notes for the 201220.

Harmonic maps into singular spaces and harmonic map is constant, and conclude that the representation has bounded image. The lipeomorphisms are the isomorphisms of this category. Morse theory for manifolds with b oundary is a p articular example. Thisyieldsthatseveralnonequivalentgeneralizationsofthe notion of harmonic function coexist in h1m,n weakly harmonic, stationary harmonic, minimizing. Let m,g be a four dimensional compact riemannian manifold with boundary and n,h be a compact riemannian manifold without boundary. Harmonic maps of complete manifolds article pdf available in discrete and continuous dynamical systems 54. Let m be a c2smooth riemannian manifold with boundary and n a com. Cohomology of harmonic forms on riemannian manifolds with boundary sylvain cappell, dennis deturck, herman gluck, and edward y.

Algorithmic and computer methods for threemanifolds. The first part of the book contains many important theorems on the regularity of minimizing harmonic maps by schoenuhlenbeck, stationary harmonic maps between riemannian manifolds in higher dimensions by evans and bethuel, and. Request pdf fbiharmonic maps between riemannian manifolds we show that if. We add a subscript to an operator or function to denote the corresponding one on n. We obtain the first and second variations of biharmonic maps on v manifolds.

Harmonic maps between finsler manifolds have been studied extensively by various researchers, see for instance, 1822. The emphasis is on some of the developed techniques in the subject and also geometric applications that followed. Pdf harmonic map heat flow with rough boundary data. Full text full text is available as a scanned copy of the original print version. Some isolation results for fharmonic maps on weighted. Harmonicfunctionson completeriemannianmanifolds peter li abstract we present a brief description of certain aspects of the theory of harmonic functions on a complete riemannian manifold.

H fz c such that jzj with the metric ds 2jdzj 1j zj2 most lines in the poincare disk model are arcs of circles that intersect the boundary s1 orthogonally. Buy algorithmic and computer methods for threemanifolds mathematics and its applications on free shipping on qualified orders. Appendix 3 deals with properties of the harmonic extension of bmo and vmo maps. We will conclude this chapter with a brief survey of the known results on weakly harmonic maps in h1m,n. These are of interest both in their own right, and to study the underlying manifold. If the answer is yes, one of the most efficient methods to prove it is to combine results on existence. Particular emphasis has been placed on the singularities\ud which may occur, as described by struwe, and the analysis of the flow despite these. Introduction to differentiable manifolds lecture notes version 2. We follow the book introduction to smooth manifolds by john m. S a, where s is a closed surface and a is a complete hyperbolic 3 manifold.

X xis lipschitz with lipidx 1 and a composition g fof lipschitz maps is lipschitz with lipg f. 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. Runtimes for the algorithm can be long because of the many harmonic map computations. Furthermore, owing to the properties of harmonic maps, harmonic shape images are able to. The theory of the energy functional and its harmonic. The universal invariant for manifolds with boundary. Lecture notes geometry of manifolds mathematics mit. Hermitian harmonic maps from complete hermitian manifolds. Singularities in the harmonic map flow with free boundary 3 proposition 1.

We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from m to n under the dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. Pdf boundary regularity and the dirichlet problem for. We present a study of the harmonic map heat flow of eells and sampson in the case that\ud the domain manifold is a surface. Harmonic mappings into manifolds with boundary numdam. A 2 harmonic map on a surface is regular up to the boundary, see hel ein hf1,2 and qing qj. In the other direction, 3manifolds have deep geometric structure, for example the. Introduction the main result of this article is the following. Harmonic maps of manifolds with boundary lecture notes in. Since a biharmonic map from a compact vmanifold into a riemannian manifold of nonpositive curvature is harmonic, we construct a biharmonic non harmonic map into a sphere. Many of our normal forms will have ambiguities in sign choices which cannot be eliminated if the source and target orientations stay.

Harmonic maps from finsler manifolds mo, xiaohuan, illinois journal of mathematics, 2001. Cohomology of harmonic forms on riemannian manifolds. Multiresolution adaptive parameterization of surfaces. The purpose of this paper is to study some submanifolds and riemannian submersions on an kenmotsu manifold. The method in ddw1 and ddw2 is based on the theory of harmonic maps from riemannian manifolds into singular spaces of nonpositive curvature npc spaces developed by gromovschoen gs, korevaarschoen ks1 ks2 ks3 and jost jo. It is not surprising that the boundary regularity is actually. Boundary regularity of stationary biharmonic maps huajun gong tobias lammy changyou wangz abstract we consider the dirichlet problem for stationary biharmonic maps ufrom a bounded, smooth domain.

The concept of kenmotsu manifold, where is a real constant, appears for the first time in. Cohomology of harmonic forms on riemannian manifolds with. Heinz on the occasion of his 80th birthday abstract on nonk. Au of these spaces are canonically homeomorphic, and we will often sup. In this article, we study the regularity of minimizing and stationary pharmonic maps between riemannian manifolds, for p. According to the general definition of manifold, a manifold of dimension 1 is a topological space which is second countable i. Harmonic maps of manifolds with boundary springerlink. Introduction generic critical value sets integer invariants mod2 invariants local order 1 invariants. Boundary behavior of harmonic maps on nonsmooth domains and.

In riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system x 1. Manifoldsandmaps uwmadison department of mathematics. Harmonic maps between rotationally symmetric manifolds volume 55 issue 3 a. Pages in category maps of manifolds the following 8 pages are in this category, out of 8 total. Lie introduced the notion of f harmonic maps between finsler manifolds.

The purpose of the present paper is to extend this theory from closed manifolds to manifolds with boundary. Received by the editors september, 2009 c 0000 american mathematical society 1. For each choice of conformai structure o and s there is a unique harmonic map in the homotopy class except in. Using a sequence of local harmonic maps, a parameterization which is smooth over each triangle in the base domain and which meets with c0 continuity at base domain edges 7, plate 1f is constructed. The harmonic map theory studies mappings between different manifolds from an energyminimization point of view. Get a printable copy pdf file of the complete article 546k, or click on a page image below to browse page by page.

With the application of harmonic maps, a surface representation called harmonic shape images is created and used for surface matching. Previously, this uniqueness result was obtained by riviere when m is the round sphere and. Harmonic selfmaps of cohomogeneity one manifolds core. Local invariants of maps between 3manifolds 759 figure 1. In section 2, we give preliminaries on kenmotsu manifolds. Boundary harmonische abbildung manifold manifolds mannigfaltigkeit. In section 4 we studied with further details the harmonic maps constructed in theorem 3. A basic example of maps between manifolds are scalarvalued functions on a manifold. Thurston the geometry and topology of threemanifolds. The reader is referred to 4 for a survey of the theory of harmonic maps. That is, given two compact riemannian manifolds m and n, where n has empty boundary, we consider the critical points of the functional epu. In a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds. Annali della scuola normale superiore di pisa, classe di scienze 4e serie, tome 17, no 3 1990, p.

It is easy to see that d is a hadamard 2 manifold of constant negative curvature 1. Available formats pdf please select a format to send. As an application, we prove a di eomorphism property for such harmonic maps in two dimensions. Harmonic maps and their applications in surface matching. We also prove that equivariant harmonic maps of finite energy from a kihler manifold into a class of riemannian simplicial complexes referred to as f.