Brendle and the secondnamed author in 3 to also include the faro. The congruence structure of conformal gradient fields on pseudo riemannian hyperquadrics and killing fields on pseudo riemannian quadrics is elucidated, and harmonic vector fields of these two types are classified up to congruence. Local gradient estimate for pharmonic functions on. In this paper, by combination of these two harmonic maps, we introduce and study fbiharmonic maps as the critical points of the fbienergy functional 1 2 r m f. M n between riemannian manifolds m and n is by definition a continuous mappings which pulls back harmonic functions. Computing harmonic maps between riemannian manifolds.
Harmonic mappings into nonnegatively curved riemannian manifolds sergey stepanov irina tsyganok abstract fifty years ago, eells and sampson have proved a famous theorem in which they argued that any harmonic mapping f. Cohomology of harmonic forms on riemannian manifolds. M at which k x is nondegenerate the proof of 1, lemma, p. We aim to give an easy to read and selfcontained introduction to the. Harmonic quasiconformal mappings of riemannian manifolds. On fbiharmonic maps between riemannian manifolds weijun lu a.
Eudml harmonic morphisms between riemannian manifolds. Harmonic morphisms are maps which preserve laplaces equation. Introduction this article surveys some developments, which started almost twenty years ago, on the applications of harmonic mappings to the study of topology and geometry of k. Albanese tori in this section, we shall consider a compact riemannian manifold m endowed with a certain measure. They can be characterized as harmonic maps which satisfy an additional first order condition.
Let m m, gti be a compact riemannian manifold of dimension m and. Hopefully the results will be useful to study corresponding rigidity of complete hermitian manifolds. Miller to julius shaneson on the occasion of his 60th birthday 1. Dirichlets boundary value problem for harmonic mappings of. Sampson in their seminal paper, harmonic mappings of riemannian manifolds 3. It is well known that the liouville theorem of har monic functions can be obtained by the harnack inequality for harmonic functions. Mn between riemannian manifolds m and n is called harmonic if it is a critical point of the dirichlet energy functional. Harmonic morphisms between riemannian manifolds paul.
M m of a compact oriented riemannian manifold m into a f complete riemannian manifold m is harmonic if its energy ef 2 vo j iijil m. V r is a harmonic function on an open subset v of c with 1v nonempty, then h. Hermitian harmonic maps from complete hermitian manifolds. Harmonic mappings and moduli spaces of riemann surfaces. About the degenerate spectrum of the tension field for mappings into a symmetric riemannian manifold kourouma, moussa, african diaspora journal of mathematics, 2017. Harmonic maps and riemannian submersions between manifolds. In the case when m and n are surfaces a complete picture is known, for example, consider the homotopy classes of maps from the. This is the first account in book form of the theory of harmonic morphisms between riemannian manifolds. Semantic scholar extracted view of harmonic mappings of riemannian manifolds by james b. An existence theorem for harmonic mappings of riemannian manifolds. The geometric properties of harmonic function on 2. Choi 6 showed a liouville type theorem for harmonic maps from a complete smooth riemannian manifold with nonnegative ricci curvature.
The theory of harmonic vector fields on riemannian manifolds is generalised to pseudoriemannian manifolds. Bourguignon has asked the question whether the ricci tensor of a compact riemannian manifold with harmonic curvature must be parallel. In conclusion, we give the applications of our results to. Local gradient estimate for pharmonic functions on riemannian manifolds xiaodong wang and lei zhang for positive pharmonic functions on riemannian manifolds, 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. Some topics in the theory of harmonic functions on. About the geometric properties of the level sets of the harmonic function defined on the space forms, mazhang derived the strict convexity and the lower bound of the gaussian curvature. Harmonic mappings into manifolds with boundary numdam. A connected riemannian manifold carries the structure of a metric space whose distance function is the arc length of a minimizing geodesic.
Introduction harmonic maps constitute a powerful tool for the analysis of moduli and teichm. Harmonic morphisms between riemannian manifolds paul baird. Pdf we investigate a the isometric and b the harmonic mappings. In section 2, we turn to the general case of harmonic morphisms between riemannian manifolds, explaining the notions of harmonic map and horizontal weak conformality.
The fundamental problem of harmonic mappings is, given a particular homotopy class of mappings between riemannian manifolds. Widman, dirichlets boundary value problem for harmonic mappings of riemannian manifolds, math. Harmonic mappings between riemannian manifolds by anand. Harmonic mappings and moduli spaces of riemann surfaces j. In section 3, we summarize some of the important constructions of. Generalization of this problem to the partial differential equations on riemannian manifolds is a natural question. Geometric formulation and main results this paper is concerned with generalizations of liouvilles and reshetnyaks theorems to the realm of mappings between riemannian manifolds. Harmonic mean curvature flow in riemannian manifolds and. Let m be a compact, connected, oriented, smooth riemannian ndimensional.
The basic question they tackled was if given a function f 2c0m. This system of equations arise as the eulerlagrange equations of a natural dirichlet or energy integral on maps between manifolds, which directly generalises the classical dirichlet integral. In section 3 we recall the notions of quaternionic. Harmonic mappings of riemannian manifolds mathematics johns. Pdf dirichlets boundary value problem for harmonic. The aim of this note is to describe an easy example answering this question in. Cohomology of harmonic forms on riemannian manifolds with boundary sylvain cappell, dennis deturck, herman gluck, and edward y. This thesis treats two topics about geometric flows. Harmonic vector fields on pseudoriemannian manifolds. Riemannian manifolds and albanese tori 679 present paper is a continuation of 8. Does every homotopy class of maps between riemannian.
Riemannian maps need not be harmonic, and harmonic maps need not. Computing discrete equivariant harmonic maps arxiv. Does every homotopy class of maps between riemannian manifolds admit a harmonic representative. Morse theory for normal geodesics in subriemannian manifolds with codimension one. U c from an open subset of rm is called a harmonic morphism if, whenever h. A harmonic endomorphism in a semiriemannian context bejan, cornelialivia and eken, semsi, 2016. This dissertation investigates some special mappings on riemannian manifolds called harmonic vector elds which have many interesting properties.
Harmonic maps heat flow convexity gradient descent. The theory of harmonic vector fields on riemannian manifolds is generalised to pseudo riemannian manifolds. Pdf regularity and quantitative gradient estimate of p. The theory works best when the metric of n has nonpositive sectional curvature. Both biharmonic map and fharmonic map have nice physical motivation. The nature of this metric leads to our consideration of what happens if we consider harmonic vector. Unstable harmonic maps into real hypersurfaces of a complex hopf manifold dragomir, sorin and enea, maria rosaria, tsukuba journal of mathematics, 2001. Harmonic mappings into nonnegatively curved riemannian. Some topics in the theory of harmonic functions on riemannian. If we apply 4 to suitably chosen local deformations e. Dirichlets boundary value problem for harmonic mappings of riemannian manifolds.
Harmonic function on riemannian manifold mathematics. Introduction let x be a compact, connected ndimensional riemannian manifold of class c a, with interior and nonvoid boundary. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a riemann surface. Harmonic function on riemannian manifold stack exchange. Both biharmonic map and fharmonic map have nice physical motivation and applications.
Cohomology of harmonic forms on riemannian manifolds with. These authors prove that if the target manifold is compact and all its sectional curvatures are nonpositive, every continuous map is. One topic concerns the deformation of hypersurfaces in negatively curved riemannian manifolds using fully nonlinear parabolic equations defined in terms of the normal curvatures. These notes originated from a series of lectures i delivered at the centre for mathematical analysis at canberra. Harmonic morphisms between semiriemannian manifolds 35 proof of lemma 2. On a liouville type theorem for harmonic maps to convex spaces via. The congruence structure of conformal gradient fields on pseudoriemannian hyperquadrics and killing fields on pseudoriemannian quadrics is elucidated, and harmonic vector fields of these two types are classified up to congruence. Jonah gaster, brice loustau, leonard monsaingeon download pdf. To fix ideas, we can assume that m is a compact riemannian manifold without boundary. Local gradient estimate for p harmonic functions on riemannian manifolds xiaodong wang and lei zhang for positive p harmonic functions on riemannian manifolds, 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. The euclidean space rn equipped with the same standard. N, there exists a harmonic function which is homotopic. M n between riemannian manifolds that minimize a certain variationalintegral,calledtheenergy. An introduction to harmonic manifolds and the lichnerowicz.
Addenda to the theory of direrentiation on a manifold 319 321 324 5. Introduction the main result of this article is the following. A riemannian metric on a manifold mis a smooth function g. An introduction to riemannian geometry with applications to mechanics and relativity leonor godinho and jos. Some topics in the theory of harmonic functions on riemannian manifolds stefano pigola bicocca, july 12, 2006 abstract in this talk we take a look at some classical and some new results on the interplay between the vanishing and the. We show that if is an fbiharmonic map from a compact riemannian manifold into a riemannian manifold with nonpositive curvature satisfying a condition, then is an fharmonic map. The rst systematic treatment of harmonic maps between manifolds of arbitrary dimension was undertaken by james eells and j.
This dissertation investigates harmonic vector elds which are special mappings on riemannian manifolds with many interesting properties. More precisely, we extend here the lyapunovschmidt analysis of outlying stable constant mean curvature spheres that developed by s. Moreover, this metric spaces natural topology agrees with the manifold s topology. M m of a compact oriented riemannian manifold m into a f complete riemannian manifold m is harmonic if its energy ef 2 vo j iijil m m. A riemannian manifold is a manifold equipped with a speci c riemannian metric. On jacobi fields along eigenmappings of the tension field for mappings into a symmetric riemannian manifold kourouma, moussa, african diaspora journal of mathematics, 2015. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with.
In our paper we develop a theory of harmonic mappings of riemannian manifolds into nonnegatively curved riemannian manifolds and give the geometric applications of these results to the theory of contraction maps of riemannian manifolds and of holomorphic maps of almost kahlerian manifolds. Harmonic mappings of riemannian manifolds springerlink. 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. Harmonic morphisms between riemannian manifolds london.
Dirichlets boundary value problem for harmonic mappings. L2harmonic forms on incomplete riemannian manifolds. An introduction to harmonic manifolds and the lichnerowicz conjecture peter kreyssig biosystems analysis group, friedrich schiller university of jena, 07743 jena, germany peter. An existence theorem for harmonic mappings of riemannian manifolds by stefan hildebrandt, helmut kaul and kjellove widman1 bonn university, germany link6ping university, sweden 1. An existence theorem for harmonic mappings of riemannian. 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. This work was partially carried out under the auspices of the sonderforschungsbereich theoretische mathematik at the university of bonn. There are mainly two ways to obtain the harnack inequality for harmonic function on complete riemannian manifold.
Regularity and quantitative gradient estimate of pharmonic mappings between riemannian manifolds article pdf available february 2018 with 52 reads how we measure reads. An introduction to differentiable manifolds and riemannian. 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. 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.
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