Download Two Reports on Harmonic Maps by James Eells PDF
By James Eells
Harmonic maps among Riemannian manifolds are options of platforms of partial differential equations which seem in several contexts of differential geometry. They comprise holomorphic maps, minimum surfaces, delta-models in physics. lately, they've got develop into robust instruments within the examine of world houses of Riemannian and Kahlerian manifolds. general references for this topic are reviews, released in 1978 and 1988 through James Eells and Luc Lemaire. This e-book provides those experiences in one quantity with a quick complement reporting on a few contemporary advancements within the conception. it's either an advent to the topic and a resource of reference, supplying an geared up exposition of effects unfold all through greater than 800 papers.
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Example text
Given J on the bundle V we extend it by C-linearity to an endomorphism Jc of V'. Since (Jc)2 = -Id, it determines eigenbundles V', V" associated with the eigenvalues ± i; and we have the direct sum decomposition Vc = V' ® V". ) There is a similar decomposition of the dual bundle (V *)c; and these induce decompositions of the various tensor product bundles. In particular, A'(V *)c = p+4=r F, VP. 9 l / with V'- ° = V' and V °' 1 = V ". 3) An Hermitian metric on (ti, J) -+ M is a Riemannian metric a such that
A holomorphic map 0: M -> P1(C) = S2 is just a meromorphic function. The Riemann-Roch theorem and the theory of Weierstrass points give much information concerning their existence. For instance, (1) there is no meromorphic function of Brouwer degree d = 1 on a Riemann surface of genus p > 1; (2) if p = 1 or 2, then such functions exist for all d > 2; (3) if p 3 3, then such functions exist for all d > p+ 1. 38 38 J. EELLS AND L. , there is a meromorphic function of degree 2), then all even degrees are represented, but no odd d < p.
5) below. For n > 3 the energy functional has infimum 0, and hence does not achieve its absolute minimum in any com- ponent of degree k 4 0 [45]. In fact, the pth harmonic suspension constructed above has Morse index > p for p = 5, 6 [206]. 5) with m = 26. We also have harmonic maps of degree ±4 for n = 15, 27, 51; and harmonic maps of all even degrees for n=9. By various such manipulations applied to the examples above we can obtain other harmonic maps between spheres representing nontrivial homotopy classes [212].