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{Harmonic Maps from Surfaces of Arbitrary Genus Into Spheres} Harmonic maps from surfaces of arbitrary genus into spheres Renan Assimos∗ Max Planck Institute for Mathematics in the Sciences Leipzig, Germany The existence of harmonic maps into a given target manifold is a central point of study in geometric analysis and has motivated several cornerstone works in the field, for instance geometric flows were introduced by J. Eells and J. Sampson [2] in 1964 to study the existence of harmonic maps into targets manifolds of non-positive curvature. In the present paper we are interested in the existence of harmonic maps with some image restrictions that arise from the classical maximum principle. We use ideas from W. Kendall in [8] to construct a smooth non-constant harmonic map from compact Riemann surfaces of arbitrary genus into S2 with no closed geodesics in their images. Such harmonic maps give a counterexample to a conjecture by M. Emery in [3]. Keywords: Harmonic maps, harmonic map heat flow, maximum principle. ∗Correspondence: [email protected] 1 1 Introduction Let (N; h) be a complete connected Riemannian manifold and let V ( N be an open connected subset of N. In addition suppose there exists a C2-function f : V −! R strictly convex, in the geodesic sense. That is, (f ◦ c)00 > 0 for every geodesic c : (−, ) −! N with c((−, )) ⊂ V . We say that such V is a convex supporting domain. Remark 1.1. Very little can be said about the geometry of a convex supporting domain n n V . For instance, consider the following examples: In Ω1 := fx 2 R j 1 ≤ jxj ≤ 2g (R 2 we can define the distance square function from the origin d (:; 0) : Ω1 −! R+, which is clearly strictly convex. On the other hand, for any given > 0 there exists no strictly n π convex function supported in Ω2 := fx 2 S jd(x; (0; 0; 1)) ≤ 2 + g. Consider the following properties on an open connected subset V ( N. (i) For V ⊂ N; there exists a striclty convex function f : V −! R (ii) There is no closed M and harmonic map non-constant φ : M ! N such that φ(M) ⊂ V (iii) There is no n ≥ 1 and harmonic map non-constant φ : Sn ! N such that φ(Sn) ⊂ V (iv) There is no harmonic map non-constant φ : S1 ! N such that φ(M) ⊂ V: By the maximum principle, one can easily see that (i) =) (ii), and (ii) =) (iii) =) (iv) are just direct consequences. In [3], M. Emery has conjectured that if a subset V ⊂ N admits no closed geodesics, that is, if it has property (iv), then there exists a strictly convex function f : V ! R; in other words, it was conjecture that (iv) implies (i). This conjecture is refuted by W. Kendall in [8], where an example of a subset of S2 without closed geodesics is given. The author proves that one cannot define a strictly convex function in that subset of S2. In the present paper we prove something slightly stronger regarding the existence of a non-constant harmonic map into the region constructed in [8]. Namely, we construct a non-constant harmonic into it. In other words, we prove that property (iv) does not imply (ii). We use the harmonic map heat flow to construct a smooth harmonic map u1 : 2 2 (Σ2; g) −! S , where Σ2 is a closed Riemann surface of genus 2, and u1 ⊂ Ω ⊂ S , 2 1 2 3 where Ω := S n (Γ ; Γ ; Γ ) is the region constructed in [8], see Figure2. It is important to remark that our construction works for Riemannian surfaces of genus bigger than 2 as well. Our construction gives a general method to constructed harmonic maps from compact Riemann surfaces of any genus into symmetric surfaces taking inspiration from 2 the existence of harmonic maps from a finite graphs. We do it in such a way that image symmetries under the harmonic map flow are preserved by the solution at time t = 1. What about the other reverse implications? In the last section of the present paper we show that the first reverse implication essentially holds true following our work [1], and add remarks and strategies to prove whether the other implications can hold true or not. A complete answer to all the implications on this diagram would be a powerful result. On one hand, it is a problem interesting on its own. On the other hand, since in [1] we have given abstract ways of showing that a region has property (ii), this would give us a conceptual way of verifying whether a region does admit a strictly convex function. A similar question to this is raised by W. Kendall in [7] but in a more Martigale flavored case. 2 Preliminaries In this section, we introduce basic notions and properties of harmonic maps to establish the necessary definitions, results and notation for the next chapters. We are closely following the books by Y. Xin [13], and F. Lin, C. Wang [10]. Let (M; g) and (N; h) be Riemannian manifolds without boundary, of dimension m and n, respectively. By Nash's theorem there exists an isometric embedding N,! RL. Definition 2.1. A map u 2 W 1;2(M; N) is called harmonic if and only if it is a critical point of the energy functional Z 1 2 E(u) := kduk dvolg (2.1) 2 M where k:k2 = h:; :i is the metric over the bundle T ∗M ⊗ u−1TN induced by g and h. Recall that the Sobolev space W 1;2(M; N) is defined as ( Z 1;2 L 2 2 2 W (M; N) = v : M −! R ; jjvjjW 1;2(M) = (jvj + kdvk ) dvolg < +1 and M ) v(x) 2 N for almost every x 2 M Remark 2.2. If u 2 W 1;2(M; N) is harmonic and φ : M −! M is a conformal diffeomorphism, i.e. φ∗g = λ2g where λ is a smooth function, we have, by (2.1), that Z 2−m 2 E(u ◦ φ) = λ kduk dvolg; (2.2) M 3 which means that the energy is a conformal invariant if m = 2. Therefore harmonic maps defined on surfaces are invariant under conformal diffeomorphisms of the domain. 1;2 −1 For u 2 W (M; N), define the map du : Γ(TM) −! u (TN), given by X 7! u∗X. We denote by rdu the gradient of du over the induced bundle T ∗M ⊗ u−1(TN), that is, ∗ −1 rdu satisfies rY du 2 Γ(T M ⊗ u (TN)), for each Y 2 Γ(TM). Definition 2.3 (Second fundamental form). The second fundamental form of the map u : M −! N is the map defined by 1 BXY (u) = (rX du)(Y ) 2 Γ(u TN): (2.3) Note that B is bilinear and symmetric in X; Y 2 Γ(TM). Indeed, we can see this map as B(u) 2 Γ(Hom(TM T M; u−1TN)): The second fundamental form will be also denoted by A(u) or rdu. Definition 2.4. The tension field of a map u : M −! N is the trace of second funda- mental form τ(u) = Beiei (u) = (rei du)(ei) (2.4) seen as a cross section of the bundle u−1TN. 1 Taking a C variation u(:; t) ktk<, it is well known that d @u E(u(:; t)) = − τ(u(x; t)); (2.5) dt @t L2 By (2.5), the Euler-Lagrange equations for the energy functional are ! @uβ @uγ @ τ(u) = ∆ uα + gijΓα h = 0: (2.6) g βγ @xi @xj βγ @uα α where the Γβγ denote the Christoffel symbols of N. Hence we can define (weakly) harmonic maps equivalently as maps that (weakly) satisfy the harmonic map equation (2.6). Definition 2.5 (Totally geodesic maps). A map u 2 W 1;2(M; N) is called totally geodesic if and only if its second fundamental form identically vanishes, i.e. B(u) ≡ 0. Another important remark on harmonic maps are the composition formulas. 4 Lemma 2.6 (Composition formulas). Let u : M −! N and f : N −! P , where (P; i) is another Riemannian manifold. Then rd(f ◦ u) = df ◦ rdu + rdf(du; du); (2.7) τ(f ◦ u) = df ◦ τ(u) + tr rdf(du; du): (2.8) Example 2.7 (L. Lemaire [9]). For every integers p and D such that jD ≤ p − 1j, there exists a Riemann surface of genus p and a harmonic nonholomorphic map of degree D from that surface to the sphere. ~ 1 For the case of D = 0, one constructs a harmonic map φ :Σg −! S and a totally geodesic embedding S1 ,! S2 such that the following diagram commutes. ~ φ:= (tot.geod) ◦ φ 2 Σg S φ~ tot. geod S1 2 2 Clearly, the image φ(Σg) ⊂ S is contained in the image of a closed geodesic of S . The main construction of this paper is of a harmonic map from a Riemann surface of even genus into S2 that does not contain closed geodesics in its image. 2.1 The harmonic map flow In 1964, J. Eells and J.H. Sampson [2] have introduced the harmonic map heat flow as away to find harmonic maps between two Riemannian manifolds. More precisely, given a 1;2 map u0 2 W (M; N), we are interested in solutions of 8 <@tu(x; t) = τ u(x; t) (2.9) :u( · ; 0) = u0: By (2.5), we have Z d E(u(x; t)) = − τ u(x; t) ;@tu(x; t) u∗TN dvolg dt M Z 2 = − τ(u(x; t)) dvolg (2.10) M 2 and since τ(u(x; t)) ≥ 0, the energy does not increase along the flow.
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