{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 := {x ∈ R | 1 ≤ |x| ≤ 2} (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 := {x ∈ S |d(x, (0, 0, 1)) ≤ 2 + }.

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 u∞ : 2 2 (Σ2, g) −→ S , where Σ2 is a closed Riemann surface of genus 2, and u∞ ⊂ Ω ⊂ S , 2 1 2 3 where Ω := S \ (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 = ∞. 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 ∈ 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 ; ||v||W 1,2(M) = (|v| + kdvk ) dvolg < +∞ and M ) v(x) ∈ N for almost every x ∈ M

Remark 2.2. If u ∈ 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 ∈ W (M,N), deﬁne the map du : Γ(TM) −→ u (TN), given by X 7→ u∗X. We denote by ∇du the gradient of du over the induced bundle T ∗M ⊗ u−1(TN), that is, ∗ −1 ∇du satisﬁes ∇Y du ∈ Γ(T M ⊗ u (TN)), for each Y ∈ 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) = (∇X du)(Y ) ∈ Γ(u TN). (2.3)

Note that B is bilinear and symmetric in X,Y ∈ Γ(TM). Indeed, we can see this map as

B(u) ∈ Γ(Hom(TM T M, u−1TN)).

The second fundamental form will be also denoted by A(u) or ∇du.

Deﬁnition 2.4. The tension ﬁeld of a map u : M −→ N is the trace of second fundamental form

τ(u) = Beiei (u) = (∇ei 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 ∈ 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

∇d(f ◦ u) = df ◦ ∇du + ∇df(du, du), (2.7)

τ(f ◦ u) = df ◦ τ(u) + tr ∇df(du, du). (2.8)

Example 2.7 (L. Lemaire [9]). For every integers p and D such that |D ≤ p − 1|, 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 ∈ W (M,N), we are interested in solutions of

 ∂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. It was proven by M. Struwe [12], in the case where the dimension of M is 2 and N is a compact manifold, that there exists a global weak solution of the heat ﬂow of harmonic

5 maps. Such a solution is smooth with the possible exception of ﬁnitely many singular points. We write down Struwe’s result together with A. Freire’s uniqueness [5],[4].

1,2 Theorem 2.8 (Struwe, Freire). For any initial value u0 ∈ W (M,N), there exists a unique global weak solution u of

 ∂tu(x, t) = τ u(x, t)

u( · , t) = u0 on M × [0, +∞) which is regular on M × (0, +∞) except for at most ﬁnitely many points (xl,T l), 1 ≤ l ≤ L characterized by the condition that

l lim sup ER u( · ,T ), x > 1 for all R ∈ (0,R0]. T →T l− On each singular point, we also have the bubbling phenomenon.

Theorem 2.9 (Struwe). Let u be the solution of

 ∂tu(x, t) = τ u(x, t)

u( · , t) = u0 constructed in the above theorem. Suppose (x0,T ), T ≤ ∞, is a point where

lim sup ER(u(., t), x0) > 1 for all R ∈ (0,R0] t→T,t

Then there exist sequences xm → x0, tm → T (tm < T ) and Rm → 0 (Rm ∈ (0,R0]) and 2 a regular harmonic map u0 : R → N such that

2,2 2 uR ,(x ,t )( · , 0) −→ u0 locally in H ( ,N) m m m m→∞ R

2 Furthermore u0 has ﬁnite energy and extends to a smooth harmonic map S → N.

The harmonic map ﬂow and Struwe’s results are the main ingredients to construct the desired harmonic map.

3 The main example

3.1 The construction of the Riemann surface

We start by considering a region on S2 described by Figure1.

6 2 Γ

1 Equator Γ3 Γ

Figure 1: A region in S2 without closed geodesics.

Notation 3.1. By ‘Equator’ we denote the embedding i : S1 ,→ S2 where i(S1) = 2 {(x1, x2, x3) ∈ S ; x3 = 0}.

Dividing the Equator into 6 equal pieces, we can pick Γ1,Γ2,Γ3 as three of those pieces that do not have pairwise intersection, like in Figure1. For a given > 0, consider the i 2 i 2 1 2 3 sets Γ := {x ∈ S | d˚g(x, Γ ) < }, for each i = 1, 2, 3. Deﬁne Ω := S \ (Γ ∪ Γ ∪ Γ ). For a point p ∈ S2 such that p ∈ Equator\(Γ1 ∪Γ2 ∪Γ3), it is obvious that its antipodal 1 2 3 point A(p) ∈ (Γ ∪ Γ ∪ Γ ). Therefore Ω does not admit closed geodesics. This is analogous to W. Kendall’s propeller in [8].

We deﬁne a compact Riemann surface Σ2 as follows. See Figure2. Consider the poles S = (0, 0, −1) and N = (0, 0, 1). Let S˜2 denote another sphere such ˜ ˜ 3 ˜2 2 that N, S belong to the axis x3 of R and S is a reﬂection of S centered at a point of the axis. Denote by d > 0 the distance between N˜ and S. In S˜2, consider also the ˜i ˜i 2 segments Γ and the sets Γ as in S . We connect the two spheres by three tubes (cylinders), called T 1, T 2, T 3 such that i i ˜i i ˜i ∂T = Γ ∪ Γ, i.e., the tubes connect the sets Γ and Γ pairwise and smoothly. It is important to stress two things about this construction.

• A Z3 symmetry around the x3 axis is preserved, i.e. the tubes and the circles are all equal.

˜ • A Z2 symmetry around the midpoint N and S is preserved.

These symmetries play an important role in the construction of the desired harmonic

7 S2

2 Γ p3 p2 T3 T1 T2 u0 smooth

1 p1 3 Γ Γ

S˜2

Figure 2: The Riemann surface Σ2 and the map u0

map. Clearly, the procedure above generates a genus 2 compact Riemann surface Σ2 and we consider the induced metric from R3 as its Riemannian metric. 2 The next step is to deﬁne the smooth initial condition between Σ2 and S and explore the properties of the harmonic map ﬂow. ∞ 2 Let u0 ∈ C Σ2,S be the map given by,  2 1 2 3  N if x ∈ S \ (C ∪ C ∪ C )  2 1 2 3 u0 : x 7→ S if x ∈ S˜ \ (C ∪ C ∪ C ) (3.1)   (∗) if x ∈ (T 1 ∪ T 2 ∪ T 3)

(∗): Consider the following geodesic connecting the south and the north poles in S2

γ :[−R, +R] −→ S2 ! π(t+R) π(t+R) t 7−→ sin 2R , 0, − cos 2R

i where 2R > 0 is the height of the tubes T in our Riemann surface Σ2. Now, given θ ∈ [0, 2π) i Γθ :[−R,R] −→ T t 7−→ (r cos θ, r sin θ, t)

8 where r is the radius of T i.

Up to the diﬀeomorphism that makes the cylinder straight, deﬁne u0|T i as a map in ∞ 2 C (Σ2,S ) such that

u0 ◦ Γθ(t) = γ(t) for every θ ∈ [0, 2π). (3.2)

It is easy to verify that the image of Σ2 under u0 has the same Z2 and Z3 symmetries of Σ2. We point out that the harmonic map ﬂow may preserve symmetries of the initial condition.

Remark 3.2. (Symmetry property of the ﬂow) Let ϕ be the Z3 action on Σ2 given by 3π the rotation by 2 over the axis connecting the poles. Let ψ be the ‘same’ Z3 action over 2 the sphere S . Obviously u0 ◦ ϕ = ψ ◦ u0, hence we have

 ∂t ψ ◦ u = τ ψ ◦ u (3.3) ψ ◦ u( · , t) = u0 ◦ ϕ or equivalently  ∂t u ◦ ϕ = τ u ◦ ϕ (3.4) u ◦ ϕ( · , t) = ψ ◦ u0. Moreover, if u is a the unique Struwe solution for (2.9) with initial condition (3.2), as ψ is obviously totally geodesic, we have

∂t(ψ ◦ u) − τ(ψ ◦ u) = dψ(u( · , t))∂tu − dψ(u( · , t))τ(u) = dψ(u( · , t)) ∂tu − τ(u) = 0 therefore ψ ◦ u is a solution to the same problem.

The same argument applies to the Z2 action (a, b, c) 7→ (a, b, −c). Therefore the 2 harmonic map u∞ :Σ2 → S given by Struwe’s solution of (2.9) with initial condition u0 is Z3 and Z2 equivariant with respect to the actions deﬁned above. Another important remark about the initial condition u0 is that we can control its energy by changing the length and radius of the tubes T i.

π Remark 3.3 (A control on the energy of u0). Since kγ˙ (t)k = 2R , we have π 0 < kdu (p)k ≤ 0 2R and therefore 1 Z π 2 0 < E(u0) ≤ dvolT i 2 T i 2R

9 π2 Z i = 2 dvolT 8R T i π3r2 = 4R

2 ˜2 By making the tubes connecting S and S in Σ2 thinner and longer, we can make the energy of u0 arbitrarily small. More precisely, given any > 0, we can pick r ∈ (0, 1) and R ∈ (1, +∞) such that E(u0) ≤ . By Equation (2.10), we know that the energy decreases along the ﬂow and therefore we have a control on the energy of ut for every t ∈ R+.

Theorem 2.8 and Theorem 2.9 above roughly tell us that given any initial condition 1,2 u0 ∈ W (M,N), there exist a regular solution to the harmonic map ﬂow with the exception of some ﬁnite points on which the energy is controlled. In other words, we have a harmonic map

u∞ : M\{q1, ..., ql} → N and around the singularities this map can be extended to a harmonic sphere h : S2 → N. Since each harmonic two-sphere which bubbles out carries some minimum amount of energy and every energy loss during the ﬂow is due to the formation of a bubble (see 2 2 [11]), we have the following. Let K : S → R be the Gauss curvature of S and K0 the maximum value of all sectional curvatures of a target manifold N. The Gauss-Bonnet theorem implies that Z 2 4π = Kdσ ≤ K0 Area of h(S ) ≤ K0E(h) S2 and therefore 4π E(h) ≥ K0 2 In our case, N is of course S , so K0 = 1. Thus 4π is the minimum amount of energy needed to form a bubble.

By the deﬁnition of the initial condition u0 and Remark 3.3, we avoid the formation of bubbles by taking Σ2 as the compact Riemann surface with tubes of length R ∈ (1, +∞) and radius r ∈ (0, 1), such that for a given 0 > 0 we have E(u0) ≤ 0 << 4π.

With this initial condition u0, we have the smooth harmonic map

2 u∞ :Σ2 → S . (3.5)

2 1 2 3 It remains to be proven that u∞(Σ2) ⊂ S \ (Γ ∪ Γ ∪ Γ ). To do so, we need the following lemma due to Courant.

10 Lemma 3.4 (Courant-Lebesgue). Let f ∈ W 1,2(D, d), E(f) ≤ C, δ < 1 and p ∈ D = R √ 2 2 {(x, y) ∈ C; x + y = 1}. Then there exists some r ∈ (δ, δ) for which f|∂B(p,r)∩D is absolutely continuous and

1 − 2 1 1 |f(x ) − f(x )| ≤ (8πC) 2 log (3.6) 1 2 δ for all x1, x2 ∈ ∂B(p, r) ∩ D.

Proof. We follow [6]. The fact that f|∂B(p,r)∩D is absolutely continuous, at least for almost every r, follows from the fact that f is in the Sobolev space W 1,2(D, Rd). Taking polar coordinates (ρ, θ) around p we have, by the intermediate value theorem, that, for all x1, x2 ∈ ∂B(p, r) ∩ D,

Z |f(x1) − f(x2)| ≤ |fθ(x)|dθ (ρ,θ)∈∂B(p,r)∩D

1 Z 2 1 2 ≤ (2π) 2 |fθ(x)| dθ

But the energy of f on B(p, r) ∩ D is

Z 1 2 1 2 E(f; B(p, r) ∩ D) = |fρ| + 2 |fθ| ρ dρ dθ 2 B(p,r)∩D ρ √ Hence, there exists a number r ∈ (δ, δ) such that Z 1 2 C 2C |fθ| dθ ≤ √ = . δ −1 2 (ρ,θ)∈∂B(p,r)∩D R −1 log(δ ) δ ρ dρ The lemma follows from these two equations.

We point out that the Z2 symmetry of u∞(Σ2) implies that there are two antipodal 3 points, called N and S, on the image of u∞(Σ2). On the other hand, the Z symmetry and the fact that the image is connected, there exist three points pi ∈ Ti, i = 1, 2, 3, each of them Z2-invariant, such that u∞(pi) ∈ Eq. Moreover, we have that ϕ(u∞(p1)) = u∞(p2), ϕ(u∞(p2)) = u∞(p3) and ϕ(u∞(p3)) = u∞(p1). Now we use the Courant-Lebesgue

3.4 to show that a small neighborhood of the tube around pi is mapped into a small r2 neighborhood of u∞(pi) of controlled size. Namely, since E(u∞) ≤ E(u0) < R , where r is the radius of the tubes and R their heights. Suppose we take r ∈ (0, 1) and R ∈ (1, +∞) such that u∞ has no bubbles and r.π << π/6.

11 Take δ < r2, by Courant-Lebesgue there exists some s ∈ (r2, r) such that

1 1 ! 2 !− 2 8π.r2 1 |u (x ) − u (x )| ≤ log (3.7) ∞ 1 ∞ 2 R δ

2 for every x1, x2 ∈ ∂Bg(pi, s) ∩ Bg(pi, 1). But since δ < r , we get

1 1 1 1 δ > s ⇒ log δ > log s

1 1 1 − 2 1 − 2 ⇒ log s > log δ . Therefore by Equation (3.7),

1 1 8π.s 2 1 − 2 |u∞(x1) − u∞(x2)| ≤ R log s

√ s q 8π π = 1 . << . 1 2 R 6 (log( s ))

1 3 In particular, if we call Si the circle given by the intersection of the x, y-plane in R 1 with Ti, the above argument shows that the image of a δ-neighborhhod of Si under u∞ is contained in a neighborhood of u∞(pi) that does not intersect any of the removed sets i 1 Γ, because since pi is invariant under the Z2 action, so is Si . 1 1 We claim for any point a ∈ Ti such that a∈ / B(Si , δ) := {x ∈ Ti | d(x, Si ) ≤ δ}, we have that u∞(a) ∈/ Equator. 2 r2 Obviously, u∞(Σ2) ( S , since Area(u∞) ≤ E(u∞) < R << 1. As Σ2 is compact, then u∞(Σ2) is compact.

If ∂u∞(Σ2) = ∅, then u∞(Ti) is totally geodesic and we have that u∞(a) ∈/ Equator 1 if a ∈ Ti \ B(Si , δ). If ∂u∞(Σ2) 6= ∅ we argue by contradiction. Suppose there exists 1 a ∈ Ti \ B(Si , δ) such that u∞(a) ∈ ∂u∞(Σ2) and u∞(a) ∈ Equator. For |t| < , consider the C1-variation

(u∞)t(x) = u∞(x) + tη(x), (3.8)

2 where η :Σ2 −→ S is smooth, η ≡ 0 outside a neighborhood of a and (u∞)t(a) ∈ 2 ˜2 HemisphereN (where we are assuming without loss of generality that d(a, S ) < d(a, S )).

This variation (u∞)t(x) clearly satisﬁes E((u∞)t) < E(u∞), but this contradicts the fact 1 that u∞ is a smooth harmonic map. Therefore such a point a ∈ Ti \ B(Si , δ) cannot be mapped on the Equator. 2 1 2 3 This concludes the proof that u∞(Σ2) ⊂ S \ (Γ ∪ Γ ∪ Γ ).

To summarize, we have shown that the solution u∞ of the harmonic map ﬂow

12  ∂tu(x, t) = τ u(x, t)

u( . , t = 0) = u0

2 where u0 :Σ2 → S is given by Deﬁnition 3.2 is a harmonic map from the compact genus 2 two Riemann surface Σ2 into the sphere S and the image u∞(Σ2) does not contain closed geodesics.

Remark 3.5. With the same argument of this example, we could construct compact Riemann surfaces of genus 2p for any p > 1. The only change in the proof would be that the Z3 symmetry would be replaced by a Z2p+1 symmetry. Moreover,if we replace the target by an other 2-dimensional surface with the appropriate symmetries, we could try to construct a u0-type initial condition respecting the necessary symmetries.

4 Remarks and open questions

As we pointed out in the introduction, a complete answer, either positive or negative, to the implications between properties (i) to (iv) of a given subset V subset(N, h) is an interesting question on its own. But we want to make a couple of remarks on this problemso that we can see some applications and possible consequences. We start by considering the equivalence between properties (i) and (ii). Since (i) implies (ii) by the maximum principle, we are considering here whether (ii) implies (i). More precisely, we are assuming that a subset V ⊂ (N, h) admits no image of non-constant harmonic maps φ :(M, g) −→ (N, h), where M is a closed Riemannian manifold, and we ask if this implies the existence of a strictly convex function f : V −→ R. A partial answer for this question was given by the authors in [1]. We present a corollary of the main tool developed in that paper with a slight modiﬁcation that is more suited for the present question. We start by remarking that if on takes a geodesic ball B(p, r) in a complete manifold N such that r is smaller than the convexity radius of N at p, then ∂B(p, r) is a hypersurface of N with deﬁnite second fundamental form for every point q ∈ ∂B(p, r). We have proven in [1] the following theorem.

Theorem 4.1. Let (N, h) be a complete Riemannian manifold and γ :[a, b] −→ N a smooth embedded curve. Consider a smooth function r :[a, b] −→ R+ and a region [ R := B(Γ(t), r(t)), (4.1) t∈[a,b]

13 where B(·, ·) is the geodesic ball and r(t) is smaller than the convexity radius of N for any t. If, for each t0 ∈ (a, b), the set R\B(γ(t0), r(t0)) is the union of two disjoint connected sets, namely the connected component of γ(a) and the one of γ(b), then there exists no compact manifold (M, g) and non-constant harmonic map φ : M −→ N such that φ(M) ⊂ R.

According to the above definitions, we can say that the region R of theorem 4.1 has property (ii). As a direct corollary of the proof, which is based on an application of a maximum principle, we have that a subset V of N that admits a sweep-out by convex hypersurfaces with the additional property that the absence of each leave of this sweep-out divides V in two connected components like the region R above, it follows that V has property (ii). This corollary gives us a ‘geometric’ way of proving that a certain subset of a manifold has property (ii) or not. Obviously, if we have f : V −→ R strictly convex, then the level sets of f naturally give us the desired sweep-out in 4.1. On the other hand, it is not true that seeing the sweep-out by convex leaves in V as level sets of a function f :−→ R, then f is strictly convex. But we can re-parameterize f so that it becomes strictly convex: It is enough to use the parameter t ∈ [a, b] of the 1-parameter family of convex leaves in the above theorem and control to growth of the gradient of f while it walks through the convex leaves of the sweep-out. The above argument gives the strategy of the proof that when V satisfies the properties of the region R in theorem 4.1, then V has also property (i). In other words, this gives us a method to obtain strictly convex functions on some subset V of (N, h) based on its geometry. Another interesting pair of possible equivalences is whether (iv) implies (iii). That is, suppose V does not have closed geodesics. Does that imply there are no harmonic maps φ :(Sk,˚g) −→ (N, h) with φ(Sk) ⊂ V ? In particular, do no closed geodesics imply no bubbles in V ? In the previous sections we have proven that property (iv) does not imply property (ii). But it can be that with similar techniques, and a different symmetric space as target (not S2), one can build up a counterexample as well to the statement (iv) implies (iii), allowing the energy of the initial condition to be big enough to form a bubble, while still controlling the image of the final map preserving symmetries, like in section 2.

14 References

[1] Renan Assimos and J¨urgenJost, The geometry of maximum principles and a Bernstein theorem in codimension 2 (2018). Submitted. Preprint available on arXiv:1811.09869. ↑3, 13

[2] J. Eells and J. H. Sampson, Harmonic mappings of riemannian manifolds, American Journal of Mathematics 86 (1964), no. 1, 109–160. ↑1,5

[3] M. Emery, Convergence des martingales dans les vari´et´es, Colloque en l’honneur de laurent schwartz (volume 2), 1985, pp. 47–63 (fr). MR816759 ↑1,2

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