By James Eells

Harmonic maps among Riemannian manifolds are ideas of structures of partial differential equations which look in numerous contexts of differential geometry. They comprise holomorphic maps, minimum surfaces, delta-models in physics. lately, they've got turn into strong instruments within the learn of world homes of Riemannian and Kahlerian manifolds. regular references for this topic are experiences, released in 1978 and 1988 by means of James Eells and Luc Lemaire. This e-book offers those reviews in one quantity with a short complement reporting on a few fresh advancements within the thought. it's either an creation to the topic and a resource of reference, supplying an prepared exposition of effects unfold all through greater than 800 papers.

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**Example text**

Example. If M is an oriented surface isometrically immersed in R", then it is minimal if and only if the Gauss map M -- Gn, 2 (the complex quadric of oriented 2-planes in R") is anti-holomorphic. Here M is endowed with the complex structure determined by the conformal class of the induced Riemann metric; see [34] and [35, 112] for further information. Example [140, 162]. Suppose that M is a compact oriented surface of genus p, isometrically and minimally immersed in a torus T. Then p = 0, 2 are impossible, and if p = 1 then M is a subtorus.

18) The following regularity theorem is an essential (and difficult) step in Uhlenbeck's proof-even for the existence of weak solutions: For all sufficiently small e > 0, the critical points of Ee in Y, 2, (M, N) are smooth maps. 19) d*(fd(D)-f Trace f (d'D, d(D) = 0, where f = 1 +eme(c)m' 1. 13). ;to be a component of S912m(M, N). A second main property is the following: 27 A REPORT ON HARMONIC MAPS 27 For all a > 0 there is a 6 > 0 such that if W is the set of all E. °. It follows easily that Y contains a harmonic map.

This equation can be interpreted as that of a pendulum with variable gravity changing sign with position. 11) has a solution a which is strictly increasing and which is asymptotic to 0 at - oo and to it/2 at + co. With that a in (8. 11) it is then shown that the first and second derivatives of 4 *+/i extend to continuous functions on R1+'-0. 3) that *,I, : Sp+r-1 - Sq+'-1 is analytic and harmonic. 11) is required to handle the cases p, r = 1. 7) to obtain harmonic maps of certain ellipsoids into spheres; and conversely.