case and

2000 Mathematics Subject Classi cation. 58E20. Key words and phrases. p-harmonic morphisms, p-harmonic maps. The second author was supported by the Regivg ;
p-HARMONIC MORPHISMS
3
where 2 C 1(M; Rq) and is the orthogonal projection of Rq unto N. In general, the regularity of weakly p-harmonic maps is not good, a fact due essentially to the non-ellipticity of the operator p at degenerate points. Away from these points, C 1 weakly p-harmonic maps are C 1 (at least for p 2). As a matter of fact, Tolksdorf T] showed that weakly p-harmonic maps need not even be C 1;1. As for many aspects of the theory, the case p = 2 is clearer: a continuous weakly harmonic map is smooth (cf. Mor, L-U, H]), but if we remove the continuity condition, all sorts of things can happen, even discontinuity at each point ( R92]). Only by imposing some extra conditions, can we expect a better regularity. For example, Takeuchi T94] showed that weakly m-harmonic maps from Rm m?1 are everywhere Holder continuous. Earlier, Duzaar had shown that into S minimising p-harmonic maps (p 2) into the closed m-hemisphere have Holder continuous rst derivatives for 2]0; 1 , if p < m < p + 2 + 2pp. The central question in the theory of p-harmonic maps is existence. Contrary to regularity, the value of p, equal or not to 2, does not play a major role, as Duzaar and Fuchs D-F] showed, by extending the crucial existence theorem of Eells and Sampson E-S]: Theorem 2.4. D-F] Let (M; g) be a compact manifold and (N; h) a compact manifold with non-positive Riemannian sectional curvature. Then, for p 2, any continuous map from M to N has, in its homotopy class, a p-harmonic representant, of minimal p-energy. When the domain manifold has dimension m, the m-energy p functional becomes conformally invariant: if g = f 2 g, then vg = f m vg , but jd jg = f1p jd jp. This ~ ~ g ~ conformal invariance implies a greater freedom of existence: Theorem 2.5. J] Let M and N be compact Riemannian manifolds. Let m = dimM and assume that m (N) = f0g. Then any 2 C 1(M; N) admits in its homotopy class an m-harmonic map of minimal m-energy. Moreover, each homotopy class of a compact Riemannian manifold possesses a p-harmonic representative: Theorem 2.6. We] If N is a compact Riemannian manifold, then for any positive integer i, each class in i(N) can be represented by a C 1; p-harmonic map from Si into N minimising p-energy in its homotopy class for any p > i. The proof of this result is rather short, if one knows that for p > i, the p-energy satis es the Palais-Smale condition (C). This implies the existence of a minimum in each component of Lp (Si; N), whose p-stability implies a C 1; -regularity ( H-L]). 1 The core of the research carried out on p-harmonic maps has been on the case p = 2 (cf. E-L78, E-L83, E-L88]). We now give an overview of some of the results speci c to p = 2: Though we have the Eells-Sampson Theorem for existence, its curvature conditions cannot be dispensed with, as is shown, for example, by: Theorem 2.7. E-W] Any harmonic non holomorphic map from the twotorus to the two-sphere is null homotopic. In particular, there is no harmonic map of degree one from T2 to S2.
Contemporary Mathematics
p-Harmonic morphisms: the 1 < p < 2
case and some non-trivial examples
J.-M. Burel and E. Loubeau
Harmonic maps, de ned as the extremums of the energy functional, nd a natural generalisation in p-harmonic maps, the critical points of the p-energy functional. Though several key properties remain valid, crucial dissimilarities in their regularity make the analysis of p-harmonic maps a delicate matter. A particular subclass of harmonic maps, the harmonic morphisms, has been found to enjoy notable geometrical attributes. Harmonic morphisms were de ned by potential theorists, as preserving local harmonic functions and characterised as horizontally weakly conformal harmonic maps by B. Fuglede F78] and T. Ishihara I] (see F99] for an overview of this theory). The temptation of de ning p-harmonic morphisms as preserving local p-harmonic functions and recovering a Fuglede-Ishihara Characterisation is the driving force behind our work (Theorem 4.3). The proof of the case p > 2 is in L2] and here we alter it to the case 1 < p < 2. From this, several well-known properties of harmonic morphisms are extended to p-harmonic morphisms, especially, a version of the Baird-Eells Theorem for p > 1 (Theorem 4.10). Finally, in the case of maps between spheres, we present a couple of examples in details. As is often the case, manifolds and metrics will be assumed smooth. In 1956 Eells, in E], showed the space of C k (0 k < +1) maps between Riemannian manifolds to be an in nite dimensional manifold. Inspired by Morse theory, he initiated the study of C k (M; N) by looking at the critical points of functionals, in connected components, i.e. within homotopy classes. Among many possible choices, the most natural seems to be the p-energy:
c 0000 (copyright holder)
1. Introduction
2. p-Harmonic maps
2
J.-M. BUREL AND E. LOUBEAU
Let : (M; g) ! (N; h) be a C 2-mapping between Riemannian manifolds. De ne, for a real number p, the p-energy of as: 1Z Ep ( ) = p jd xjp vg : M (or on any compact subset K M). Other potential candidates were the exponential energy:
E(
)=
Z
1 where e( ) = 2 jd xj2, or the biharmonic energy: Z 2 2( ) = 1 E 2 M j 2( )j vg ; (see further down for the de nition of 2 ( )). The most prosperous work has been on the p-harmonic energy, so we will concentrate on it. First, we must notice that for p < 1, Ep is not a norm and W 1;p is not a Banach space. Besides, for p = 1, while W 1;1 becomes a Banach space, it is impossible to derive a Euler-Lagrange equation corresponding to critical points of the 1-energy. Therefore, from now on, we will assume p > 1. Definition 2.1. A C 2-map is called a p-harmonic map if it is a critical point, in its homotopy class, of Ep (or Ep(K) for all compact subsets K). The Euler-Lagrange equation associated to extremums of Ep is the p-tension eld: p ( ) =jd jp?4 jd j2tracerd + p?2 d grad(jd j2) 2 Theorem 2.2. A map is p-harmonic if and only if p ( ) = 0. The case of p = 2, the harmonic maps, generalises harmonic functions and geodesics. In fact, p-harmonic maps can be transformed in harmonic maps: Lemma 2.3. B-G] Let : (M m ; g) ! (N; h) (m 3) be a C 1 -map with no degenerate point. Then is p-harmonic if and only if : (M m ; g) ! (N; h) is ~ 2(p?2) harmonic with g = jd j m?2 g. ~ Assuming N to be isometrically embedded in Rq, by Nash's Theorem for example, there exists a divergence form for p : ? p ( ) = div jd jp?2d ? jd jp?2 (d ; d ); where is the second fundamental form of the isometric embedding of N into Rq. If 1 < p < 2, then the possible existence of degenerate points, i.e. such that jd (x)j = 0, implies that p might exist everywhere only in a weak sense. Moreover, it is possible to make sense of weakly p-harmonic maps, by considering W 1;p maps and requiring either that they satisfy p = 0 in a distributional sense or, since we cannot consider homotopy classes anymore, that they are extremums of the p-energy functional for all variations of the type: ut (x) = (u(x) + t (x)) 8t 2] ? ; + ;