This dissertation investigates harmonic vector elds which are special mappings on riemannian manifolds with many interesting properties. More information on harmonic maps can be found in the following articles and books. Jun 26, 2008 polynomial growth harmonic functions on complete riemannian manifolds lee, yong hah, revista matematica iberoamericana, 2004. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with. Let m and n be compact riemannian manifolds with the sectional curvatures. On fbiharmonic maps between riemannian manifolds weijun lu a. An existence theorem for harmonic mappings of riemannian. A riemannian manifold is an abstraction of a curved space in which it is possible to measure geometric quantities such as length of a curve segment, the area of. By reducing the assumption on the ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general.
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. The study of harmonic maps is an old problem which occupied generations of mathematicians. Generalization of this problem to the partial differential equations on riemannian manifolds is a natural question. Harmonic morphisms are maps which preserve laplaces equation. A riemannian metric g on m is a smooth family of inner products on the tangent spaces of m.
An introduction to harmonic manifolds and the lichnerowicz conjecture peter kreyssig biosystems analysis group, friedrich schiller university of jena, 07743 jena, germany peter. Any analytic ndimensional riemannian manifold admits on analytic local isometric embedding in rsn. If we apply 4 to suitably chosen local deformations e. To overcome these difficulties, we will combine some ideas from. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a riemann surface. Harmonic mappings between riemannian manifolds by anand. About the degenerate spectrum of the tension field for mappings into a symmetric riemannian manifold kourouma, moussa, african diaspora journal of mathematics, 2017 a harmonic endomorphism in a semiriemannian context bejan, cornelialivia and eken, semsi, 2016. With any smooth mapping of one riemannian manifold into another it is possible to associate a variety of invariantly defined func. Browse other questions tagged riemanniangeometry or ask your own question. Uniqueness of harmonic mappings and of solutions of. Math 6397 riemannian geometry,hodge theory on riemannian manifolds by min ru, university of houston 1 hodge theory on riemannian manifolds global inner product for di.
Recently there is an increasing interest in the study of harmonic functions and harmonic mappings on nonsmooth spaces. It is well known there is no nonconstant harmonic map from a closed riemannian manifold of positive ricci curvature to a complete riemannian manifold with nonpositive sectional curvature. On the evolution of harmonic mappings of riemannian surfaces michael struwe 1 commentarii mathematici helvetici volume 60, pages 558 581 1985 cite this article. Harmonic vector fields on pseudoriemannian manifolds. 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. Conversely, when we have some mapping into m p or t p, not necessarily holomorphic, we can appeal to general existence theorems about harmonic maps to deform it into a harmonic map. Some topics in the theory of harmonic functions on riemannian. Here we quote without proof the following important theorem ms39. We aim to give an easy to read and selfcontained introduction to the.
A riemannian metric tensor makes it possible to define several geometric notions on a riemannian manifold, such as angle at an intersection, length of a curve, area of a surface and higherdimensional analogues volume, etc. This is really one of the great insights of riemann, namely, the separation between the concepts of space and metric. Harmonic morphisms between semiriemannian manifolds 35 proof of lemma 2. The nature of this metric leads to our consideration of what happens if we consider harmonic vector. Riemannian manifolds are thus a specialisation of pseudo riemannian manifolds, for which we demand that at every p2mg p is positive in the sense that g px p. Harmonic maps from closed riemannian manifolds with. To fix ideas, we can assume that m is a compact riemannian manifold without boundary, and. Harmonic mappings into manifolds with boundary numdam. Quasiconformal harmonic maps into negatively curved manifolds donnelly, harold, illinois journal of mathematics, 2001. M n between riemannian manifolds m and n is by definition a continuous mappings which pulls back harmonic functions. About the degenerate spectrum of the tension field for mappings into a symmetric riemannian manifold kourouma, moussa, african diaspora journal of mathematics, 2017. Introduction let x be a compact, connected ndimensional riemannian manifold of class c a, with interior and nonvoid boundary.
Recently the corresponding notions for semiriemannian manifolds have been studied by parmar 12. In par ticular, we obtain as a corollary a recent result of hardtwolf on. Chapter 3 riemannian manifolds i the subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in riemannian manifolds. A course in riemannian geometry trinity college dublin. In this paper, we study the behavior of weakly harmonic functions on smooth manifolds with lipschitz riemannian metrics. Harmonic mappings of compact riemannian manifolds to nonnegatively curved riemannian manifolds in this section, we will show the main theorem of our work, which we have proved by the bochner technique. Vanishing theorems for projective and harmonic mappings. Here, again, we need to use some negative curvature. Browse other questions tagged riemannian geometry or ask your own question. Harmonic morphisms between riemannian manifolds paul baird. Uniqueness of harmonic mappings and of solutions of elliptic equations on riemannian manifolds. These notes originated from a series of lectures i delivered at the centre for mathematical analysis at canberra. P harmonic mappings between riemannian manifolds changyu guo and changlin xiang abstract. Does every homotopy class of maps between riemannian.
Introduction to riemannian manifolds all manifolds will be connected, hausdor. 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. Following deane yangs sketch, we give the first detailed proof of the shorttime existence of the local ricci flow introduced by him. Harmonic functions on manifolds 727 reverse poincare inequality. The geometric properties of harmonic function on 2. An existence theorem for harmonic mappings of riemannian manifolds. Pdf harmonic mappings of riemannian manifolds semantic. Eells then wrote one of the earliest books about harmonic maps which was published in 1980, selected topics in harmonic maps with l. Suppose that m is a not necessarily compact smooth manifold.
X p 0 8x p2 t pm, that is, g p is an inner product on t pm. N between riemannian manifolds is called harmonic if it is a critical point of the energy functional e z m d2. Harmonic function on riemannian manifold mathematics stack. Riemannian manifolds and albanese tori 679 present paper is a continuation of 8. Both biharmonic map and fharmonic map have nice physical motivation and applications. Harmonic morphisms are mappings between riemannian manifolds which preserve laplaces equation. If r 1,u is harmonic on mn, p e m, and r 0, then there exists cr crr 0. Harmonic function on riemannian manifold mathematics. M n be a nonconstant harmonic map from closed riemannian manifold m to a complete riemannian manifold n. To combine the statements for vertical and horizontal parts, set.
An introduction to harmonic manifolds and the lichnerowicz. The other topic concerns ricci flow on noncompact manifolds. Let m be a c2smooth riemannian manifold with boundary and n a complete c2smooth riemannian manifold. Riemannian manifold equipped with a real analytic isometric embedding into a euclidean space. Harmonic mappings and moduli spaces of riemann surfaces. Let m be a c2smooth riemannian manifold with boundary and n a com. 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. More precisely, we extend here the lyapunovschmidt analysis of outlying stable constant mean curvature spheres that developed by s. Full text of on fharmonic morphisms between riemannian. An existence theorem for harmonic mappings of riemannian manifolds by stefan hildebrandt, helmut kaul and kjellove widman1 bonn university, germany link6ping university, sweden 1. Harmonic morphisms between riemannian manifolds paul.
A common convention is to take g to be smooth, which means that for any smooth coordinate chart u,x on m, the n 2 functions. V r is a harmonic function on an open subset v of c with 1v nonempty, then h. M n is a smooth map between smooth manifolds, denote the associated map on txm by dfx. In section 3, we summarize some of the important constructions of. Uniqueness of harmonic mappings and of solutions of elliptic. The overflow blog defending yourself against coronavirus scams. Eudml harmonic morphisms between riemannian manifolds. 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. Harmonic mean curvature flow in riemannian manifolds and.
On the evolution of harmonic mappings of riemannian surfaces. In the last section, we discuss the global isometric embedding to smooth riemann manifolds. This is the first account in book form of the theory of harmonic morphisms between riemannian manifolds. Regularity theory for pharmonic mappings between riemannian. Brendle and the secondnamed author in 3 to also include the faro. In differential geometry, a riemannian manifold or riemannian space m, g is a real, smooth manifold m equipped with a positivedefinite inner product g p on the tangent space t p m at each point p. Some topics in the theory of harmonic functions on. The main results of the theory of harmonic maps, in the large, are the results about harmonic mappings with values in riemannian manifolds of nonpositive sectional curvature based on an.
Relative energy gap for harmonic maps of riemann surfaces. Semantic scholar extracted view of harmonic mappings of riemannian manifolds by james b. Mn between riemannian manifolds m and n is called harmonic if it is a critical point of the dirichlet energy functional. Isometric or harmonic mappings of complete riemannian. A harmonic endomorphism in a semiriemannian context bejan, cornelialivia and eken, semsi, 2016. The aim of this article is to underline several properties of harmonic maps on riemannian manifolds. They can be characterized as harmonic maps which satisfy an additional first order condition. M n, whose image lies in a compact subset of n, is locally c1.
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 mappings into nonnegatively curved manifolds. Note that the set of all isometries of a riemannian manifold m,g forms a group, called the isometry group of m,g, with respect to the operation of composition of mappings, which we will denote by isomm,g. Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. Browse other questions tagged pde riemannian geometry harmonic analysis harmonic functions ellipticequations or ask your own question. We investigate a the isometric and b the harmonic mappings. The extension of the notion of harmonic map to the semiriemannian case is. Harmonic morphisms between riemannian manifolds london. Sampsons paper called harmonic mappings of riemannian manifolds published in 1964. The theory of the energy functional and its harmonic. Does every homotopy class of maps between riemannian manifolds. Using the local ricci flow, we prove the shorttime existence of the ricci flow on noncompact manifolds whose ricci curvature has a global lower. Morse theory for normal geodesics in subriemannian manifolds with codimension one.
Harmonic mappings of riemannian manifolds johns hopkins. The study of riemannian manifolds constitutes the subject called riemannian geometry. I am wondering if this remains true for other riemannian manifolds. M at which k x is nondegenerate the proof of 1, lemma, p. U c from an open subset of rm is called a harmonic morphism if, whenever h. 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. In conclusion, we give the applications of our results to. Some constructions of harmonic maps, 50 years after eells. Pdf regularity and quantitative gradient estimate of p. They can be characterized as harmonic maps which enjoy an extra property called horizontal weak.
Let m m, gti be a compact riemannian manifold of dimension m and. Regularity and quantitative gradient estimate of pharmonic mappings between riemannian manifolds article pdf available february 2018 with 52 reads how we measure reads. 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. Vanishing theorems for harmonic mappings into nonnegatively.
We refer to oneill 11 concerning semiriemannian manifolds where the metric tensor may be inde. Harmonic maps on riemannian manifolds livia tanase abstract. Albanese tori in this section, we shall consider a compact riemannian manifold m endowed with a certain measure. One of the central questions in the theory of harmonic maps is. This has been followed by various papers and books on di erential geometry and. Harmonic maps from closed riemannian manifolds with positive scalar curvature.