If we apply 4 to suitably chosen local deformations e. Introduction this paper provides a step towards a classi. Computing harmonic maps between riemannian manifolds iecl. Semantic scholar extracted view of harmonic mappings of riemannian manifolds by james b. M n 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. Uniqueness of harmonic mappings and of solutions of elliptic equations on riemannian manifolds. The key idea is to associate the histogram of a color image to a riemannian manifold. Let x be a compact, connected ndimensional riemannian manifold of class c a. In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and. One of the consequences is that for a harmonic map from a space with a nonnegative ricci tensor to a space of nonpositive sectional curvatures, the map is totally geodesic, with constant energy density, and if the domain has any point with positive ricci curvature, then the map. M at which k x is nondegenerate the proof of 1, lemma, p. A survey on differential geometry of riemannian maps. Hildebrandt, harmonic mappings of riemannian manifolds. We study in this paper harmonic maps and harmonic morphisms on kenmotsu manifolds.
In this approach the emphasis is on a correspondence between the solution of the einstein field equations and the geodesics in an appropriate riemannian configuration space. Computing harmonic maps between riemannian manifolds. Harmonic mappings and minimal immersions, montecatini, 1984, 1117, lecture notes in math. Using hamiltonjacobi techniques, we obtain the geodesics and construct the resulting. With any smooth mapping of one riemannian manifold into another it is possible to associate a variety. Compact riemannian manifold an overview sciencedirect. These notes originated from a series of lectures i delivered at the centre for mathematical analysis at canberra. Harmonic mappings into nonnegatively curved manifolds.
Recent developments extend much of the known theory of classical harmonic maps between smooth riemannian manifolds to the case when the target is a metric space of curvature bounded from above. On harmonic and pseudoharmonic maps from pseudohermitian manifolds tian chong, yuxin dong, yibin ren and guilin yang abstract. Sobolev mappings between manifolds and metric spaces. In our paper we develop a theory of harmonic mappings into riemannian manifolds with nonnegative sectional curvature. In the present paper we shall use the direct mcthod of the calculus of variations to con struct a weak solution u of the problem eu min with the side conditions ur b and ux mq, where m r 2ff. Gradient estimate for exponentially harmonic functions on complete riemannian manifolds jiaxian wu, qihua ruan, and yihu yangy abstract the notion of exponentially harmonic maps was introduced by j. Harmonic maps are extremals of a natural energy integral. In this context, the energy of the matching between the two images is measured by the dirichlet energy of the mapping between the riemannian manifolds. The main aim of this paper is to state recent results in riemannian geometry obtained by the existence of a riemannian map between riemannian manifolds and to introduce certain geometric objects along such maps which allow one to use the techniques of submanifolds or riemannian submersions for riemannian maps. In riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system x 1. With any smooth mapping of one riemannian manifold into another it is possible to associate a variety of invariantly defined func. Local gradient estimate for p harmonic functions on riemannian manif olds xiaodong wang and lei zhang for positive p harmonic functions on riemannian manif olds, we derive a gradient estimate and harnack inequality with constants depending only on the lower bound of the ricci curvature, the dimension n, p and the radius of the ball on which the func. Uniqueness of harmonic mappings and of solutions of. Harmonic mappings between riemannian manifolds oxford.
Ricci soliton iff the metric g is a solution of the nonlinear stationary pdf. Recently hang and lin provided a complete solution to this problem. This system is more appropriate to hermitian geometry than the harmonic map system since it is compatible with the holomorphic structure of the domain manifold in the sense that holomorphic maps are hermitian harmonic maps. Harmonic mappings into nonnegatively curved riemannian. Gradient estimate for exponentially harmonic functions on. This dissertation investigates some special mappings on riemannian manifolds called harmonic vector elds which have many interesting properties. Now we call a harmonic map cf from a compact riemannian manifold without boundary stable if the index of cf is zero ct. Some foliated results, pluriharmonicity and siusampson type results are established for both harmonic maps and pseudoharmonic maps. Let m,g and n,h be m and n dimensional riemannian manifolds, and let u denote a smooth map. Riemannian maps need not be harmonic, and harmonic maps need not. Harmonic quasiconformal mappings of riemannian manifolds. A bochner formula for harmonic maps into nonpositively.
Moreover, the second aim of this paper is to give an application of the stability of a harmonic map. In this paper, we study the behavior of weakly harmonic functions on smooth manifolds with lipschitz riemannian metrics. Sobolev mappings between manifolds and metric spaces piotr haj. Some regularity results for pharmonic mappings between. The formula of the title is computed, and is used to calculate the index and nullity in several cases. Dirichlets boundary value problem for harmonic mappings of riemannian manifolds. On harmonic field in riemannian manifold mogi, isamu, kodai mathematical seminar reports, 1950 the harmonic field of a riemannian manifold halperin, steve, journal of differential geometry, 2014 geodesic mappings onto riemannian manifolds and differentiability hinterleitner, irena and. Lecture 1 notes on geometry of manifolds lecture 1 thu. Recently there is an increasing interest in the study of harmonic functions and harmonic mappings on nonsmooth spaces. Hermitian harmonic maps from complete hermitian manifolds.
V r is a harmonic function on an open subset v of c with 1v nonempty, then h. This chapter is devoted to the description of those properties of harmonic maps, which are essential to the development. Pdf dirichlets boundary value problem for harmonic. Harmonic mappings into nonnegatively curved manifolds sergey stepanov irina tsyganok abstract so far, all known results on harmonic maps between riemannian manifolds are based in an essential way on the assumption that the target manifold has nonpositive sectional curvature. Suppose that f is a c 2 diffeomorphism of a compact riemannian manifold m preserving a probability measure. In this note, our aim is to enrich some regularity results in this respect, particularly in the case 1 harmonic function on a complete riemannian manifold of nonnegative ricci curvature is constant. We will study stable harmonic maps to determine what kind of harmonic maps are stable.
In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudohermitian manifolds into riemannian manifolds or k ahler manifolds. The purpose of the lectures was to introduce mathematicians familiar with the basic notions and results of linear elliptic partial differential equations and riemannian geometry to the subject of harmonic mappings. You have to spend a lot of time on basics about manifolds, tensors, etc. Widman, dirichlets boundary value problem for harmonic mappings of riemannian manifolds, math. An existence theorem for harmonic mappings of riemannian. This paper contains a new criterion for mappings of riemannian manifolds to be harmonic and as a consequence new proofs of earlier results on harmonic mappings of closed compact riemann surfaces. A harmonic morphism between arbitrary riemannian manifolds is a type of harmonic map. Isometric or harmonic mappings of complete riemannian. Harmonic mappings of riemannian manifolds and stationary. An existence theorem for harmonic mappings of riemannian manifolds. U c from an open subset of rm is called a harmonic morphism if, whenever h. Harmonic morphisms between semi riemannian manifolds 35 proof of lemma 2. Yuri kifer, peidong liu, in handbook of dynamical systems, 2006. Harmonic mappings of riemannian manifolds springerlink.
Dirichlets boundary value problem for harmonic mappings. The aim of this paper is to relate, in the case of riemannian polyhedra, the theory of harmonic maps and morphisms developed by eellsfuglede in 11, to the notion of brownian motions in riemannian polyhedra see 4, in order to generalize darlings results see 8 or 19 for the smooth case. The classical concept of geodesic and the new concept of convex concave curve on a riemannian. Harmonic mappings and moduli spaces of riemann surfaces. M at which k x is degenerate it appears from 1 that 3 holds at xwith. Moreover, other main results of the theory of harmonic mappings in the large are the results on harmonic maps into nonpositively curved riemannian manifolds. Schoen lp and mean value properties of subharmonic functions on riemannian manifolds. Local gradient estimate for pharmonic functions on. Eells and sampson, which states that any given map from a riemannian manifold to a. We will follow the textbook riemannian geometry by do carmo.
Harmonic mappings of riemannian manifolds mathematics johns. It then focuses on several examples of harmonic vector elds, such as harmonic unit vector elds, the hopf vector eld and. Widman, on the h older continuity of weak solutions of. Uniqueness theorems for harmonic maps into metric spaces.
System upgrade on feb 12th during this period, ecommerce and registration of new users may not be available for up to 12 hours. Further applications of harmonic mappings of riemannian. Pdf harmonic mappings of riemannian manifolds semantic. Let m,g and n,h be m and n dimensional riemannian manifolds, and let u denote a smooth map from m to n, i. Regularity theory for pharmonic mappings between riemannian manifolds have been explored extensively in the literature, see subsection. The variational characterization of a harmonic mapping of riemannian manifolds f. Harmonic mappings between riemannian manifolds by anand. We also give some results on the spectral theory of a harmonic map for which the target manifold is a kenmotsu manifold. Regularity theory for p harmonic mappings between riemannian manifolds have been explored extensively in the literature, see section 1. Uniqueness of l1 solutions for the laplace equation and the heat equation on riemannian manifolds.
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