By James Eells
Harmonic maps among Riemannian manifolds are ideas 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 turn into robust instruments within the examine of worldwide homes of Riemannian and Kahlerian manifolds. regular references for this topic are stories, released in 1978 and 1988 by way of James Eells and Luc Lemaire. This ebook provides those experiences in one quantity with a short complement reporting on a few contemporary advancements within the thought. it's either an advent to the topic and a resource of reference, offering an equipped exposition of effects unfold all through greater than 800 papers.
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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. If M is not hyperelliptic, then there is a meromorphic function with d = p. In general, there is a meromorphic function from some d < [(p+3)/2].
Indeed, the above expression for Ae'(¢) and the condition e'(-O)-e"(0) = I imply that a"(c)) is constant. 5) below insures that e"(0) has zeros, so that e"(4)) = 0; then e(4)) = 1. 25) yields this conclusion: Let M, N be Kuhler manifolds with RicciM > 0, HB RiemN < 0; assume that M is compact. Then any holomorphic map 0: M -+ N is totally geodesic. Furthermore, if Ricci' > 0 at some point, then 0 is constant. 10. Harmonic maps of a surface (10. 1) In this section we are concerned primarily with harmonic maps 0 : (M, g) --+ (N, h), where M is compact and dim M = 2; we shall also assume that dim N >, 2, for simplicity.
34 J. EELLS AND L. LEMAIRE 34 Then the join 0 * 0 can be deformed into a harmonic map S1 +r-1 and p = r, then the damping conditions are unnecessary. 8) Example. 3) for any degree k > 0. Then for p < 7 the damping conditions are satisfied. The theorem implies that the p-fold suspension 0 * 0 : Sp+1 -* Sp+1 has degree k and is homotopic to a harmonic map. ) Otherwise said, every element of the homotopy group Z has a harmonic representative for n < 7. 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].