Morse Index Bound for Minimal Two Spheres

Given a closed manifold of dimension at least three, with non trivial homotopy group \pi_3(M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bound one, with sum of their energies realizes a geometric invariant width.


Introduction
Finite-dimensional Morse theory was developed by Morse [Mil63] to study geodesics. Index of a critical point of a proper nonnegative function on a manifold reflects its topology. A natural extension of Morse theory of closed geodesics would be a Morse theory of harmonic surfaces in a Riemannian manifold. Sacks and Uhlenbeck introduced α-energy [SU81], which can be perturbed to be Morse functions. The α-energy approaches the usual energy as the parameter α in the perturbation goes to one, and the corresponding critical point of α-energy converges to a harmonic map. However, without curvature assumption [LLW17] or finite fundamental group [CT99] for the ambient manifold (M, g), the harmonic sphere constructed by α-energy fails to realize the energy as α goes to one [LW15][Moo17, Remark 4.9.6]. Thus, we are motivated to prove the Morse index bound of the harmonic sphere produced by the min-max theory [CM08], which rules out the energy loss, namely: Theorem 1.1 (Main Theorem). Let (M, g) be a closed Riemannian manifold with dimension at least three, g generic and a nontrivial homotopy group π 3 (M ). Then there exists a collection of finitely many harmonic spheres {u i } n i=0 , u i : S 2 → M , which satisfies the following properties: (1) The collection of finitely many harmonic spheres in Theorem 1.1 is constructed by Colding and Minicozzi [CM08] by the min-max theory for the energy functional, which they used to prove finite extinction time of Ricci flow. The theory can be loosely spoken as the following: given a closed manifold M , sweep M out by a continuous one parameter family of maps from S 2 to M , starting and ending at point maps. Pull the sweepout (Definition 2.16) tight, in a continuous way, by harmonic replacements. Then if the sweepout induces a nontrivial class in π 3 (M ), then each map in the tightened sweepout whose area is close to the width must itself be close to a finite collection of harmonic spheres, close in the bubble tree sense (Definition 2.9). In other words, the width is realized by the sum of areas of finitely many harmonic spheres. Theorem 1.1 states that the sum of Morse indices of the harmonic spheres is at most one.
On the other hand, harmonic spheres are minimal surfaces. Almgren and Pitts' minmax theory [Pit81] proves the existence of embedded minimal hypersurfaces in closed manifold of dimension at least three at most seven. Marques and Neves proved the Morse index bound of such an embedded minimal hypersurface [MN16]. This result plays an important role in proving Yau's conjecture [Son18], which states that for any closed three manifold there exist infinitely many embedded minimal surfaces. However, Almgren and Pitts' min-max theory doesn't say anything about minimal surfaces in higher codimension. While using the min-max theory of harmonic spheres by Colding and Minicozzi [CM08], there's no restriction on codimension of the ambient manifold. So it motivates us to prove the Morse index bound of the harmonic sphere produced by the min-max theory of Colding and Minicozzi [CM08].
Theorem 1.1 seems like a variant of [MN16]. We compare the difference between them here. Besides the obvious difference of harmonic spheres and embedded minimal hypersurfaces, codimension restriction of ambient manifold, the embedded minimal hypersurface used in [MN16] is given by Almgren-Pitts' min-max theory, thus could have several components. When considering the variation of it, it means the variation of the whole configuration instead of each component. The index of [MN16] is the maximal dimension on which the second variation of the area functional of the whole configuration is negative definite. (But since the components are disconnected embedded minimal hypersurfaces, the index of the whole configuration is equivalent to the sum of Morse indices of each component.) While in theorem 1.1, the finite collection of harmonic spheres are not necessarily disconnected, the index of the whole configuration is less than or equal to the Morse index sum of each harmonic sphere. But the index bound we obtain in theorem 1.1 is the sum of Morse indices of each component, which is stronger than the bound for the whole configuration.
We also mention the following Morse Index conjecture proposed by Marques and Neves [MN18]: Morse Index Conjection For generic metric on M n+1 , 3 ≤ (n+1) ≤ 7, there exists a smooth, embedded, two-sided and closed minimal hypersurface Σ such that Index(Σ) = k for any integer k. Marques and Neves have shown that the conjecture is true under the assumption of multiplicity one [MN18]. For harmonic spheres, we consider the case of same assumption in theorem 1.1, the conjecture is true if all the finite collection of harmonic spheres in the image set (see definition 2.21) is one harmonic sphere. In other words, if the min-max sequence (see definition 2.18) converges to only one harmonic sphere strongly, then that harmonic sphere has Morse index one. For the general case, the difficulty lies in bubble convergence (definition 2.7). Ideally, we want to use the idea that a local minimizer can't be a min-max limit, and a stable harmonic sphere is a local minimizer for energy functional among all the spheres lie in the small tubular neighborhood. But bubble convergence doesn't imply the min-max sequence lies in the tubular neighborhood of one harmonic sphere, thus making it hard to conclude that the harmonic spheres in the image set can't all be stable.
1.1. Idea of the Proof for Theorem 1.1. We consider the image set Λ({γ j (·, t)} j ) of a minimizing sequence {γ j (·, t)} j∈N . The idea is if Index(u i ) > 1, then we are able to perturb {γ j (·, t)} j to a new sweepout {γ j (·, t)} j such that it is homotopic to γ j , it is a minimizing sequence, and , then we can perturb {γ j (·, t)} j again and get a new sweepout such that neither {u i } m i=0 nor {v i } m 0 i=0 is in its image set. Proposition B.25 states that the set of harmonic spheres with bounded energy W is countable, which allows us to perturb the sweepout inductively and get a sweepout which is away from all harmonic spheres whose sum of Morse indices is greater than 1. Since it is a minimizing sequence, it converges to a collection of finitely many harmonic spheres whose sum of Morse indices is bounded by one.
Before constructing {γ j (·, t)} j , we define the variation of a map. Suppose M is isometrically embedded in R N and let Π : R N → M be the nearest point projection from R N to M . Given a map u : S 2 → M and X : S 2 → R N , with each X i ∈ C ∞ (S 2 ), we consider the variation of u with respect to X to be u s := Π • (u + sX). We choose to define the variation this way so that for any map v : S 2 → M close to u in W 1,2 (S 2 , M ), the variation v s is close to u s as well. Assume Index(u i ) = k ≥ 2, then there exists {X l } k l=1 , X l : S 2 → R N , with the following property: for each X l , there exists at least one u l ∈ {u i } m i=0 so the second variation of energy of u l with respect to X l is negative. The idea is using {X l } k l=1 to perturb γ j (·, t). We first prove in lemma 3.1 that for γ j (·, t) close to {u i } m i=0 , there exist corresponding cutoff functions η j l : S 2 → R. LetX l := η j l X l and define the variation of γ j (·, t) with respect toX l to be: here s = (s 1 , ..., s k ) ∈B k ,B k is the k-dimensional unit ball, so that the energy of γ j,s (·, t) is concave while changing s ∈B k . That is, define E t j (s) := E(γ j,s (·, t)), E t j :B k → R, and we have D 2 E t j (s) < 0, ∀s ∈B k . If we can construct a continuous function s j : [0, 1] →B k so that energy decreases by a certain amount when γ j (·, t) is close to {u i } m i=0 . Then the sequence {γ j,s j (t) (·, t j )} j does not converge to {u i } m i=0 , and γ j,s j (t) (·, t) is the desired sweepout. In order to construct s j (t), we observe the following one parameter gradient flow {φ t j (·, x)} ∈ Diff(B k ), with x ∈B k as starting point, generated by the vector field: x) decreases the energy, except when |x| = 1 or x is the maximal point of E t j . The assumption of the lower bound of Morse index m i=0 Index(u i ) = k > 1, now enables us to construct a continuous curve y j : [0, 1] →B k avoiding the maximal point of the function E t j as t varies. Namely, let ∇E t j (x(t)) = 0, x j : [0, 1] →B k , x(t) is a continuous curve onB k . Since the dimension ofB k is larger than 1, we can choose a continuous curve y j (t) onB k which does not intersect with x j ([0, 1]), then we can use {φ t j (·, y j (t))} to construct s j (t) and obtain the new sweepoutγ j (·, t) := γ j,s j (t) (·, t). The new sweepout is homotopic to {γ j (·, t)} j , and doesn't bubble converge to {u i } m i=0 . This is the desired perturbed sweepout.
The organization of the paper is as follows. In section 2 we give the basic definitions of harmonic sphere, bubble convergence, and state the min-max theorem 2.22. In section 3 we prove a technical lemma 3.1. In section 4 we prove the main result theorem 1.1. Neves. This work would not have been possible without his generous support and insightful guidance. I would like thank Xin Zhou for many valuable discussions and inspiring suggestions.

Background Material
2.1. Harmonic Sphere. Suppose that S 2 is a Riemann sphere, which can be regarded as C ∪ {∞}, and M is a closed manifold of dimension at least three, isometrically embedded in R N .
We introduce nearest point projection Π : R N → M which maps a point x ∈ R N to the nearest point of M . There is a tubular neighborhood of M on which Π is well-defined and smooth. For a map u : S 2 → M ⊆ R N , u = (u 1 , u 2 , .., u N ), and u ∈ W 1,2 (S 2 , M ), we write ∇u as the sum of gradient of u i for i = 1, ..., N . That is, ∇u := N i=1 ∇u i , and energy of u is simply For a given X ∈ C ∞ (S 2 , R N ), we consider the variation of u with respect to X defined as the following: (2) u s = Π • (u + sX), u s is well-defined for s small enough such that the image of u + sX is in the tubular neighborhood M δ .
Definition 2.1 (Harmonic Sphere). We say that u ∈ W 1,2 (S 2 , M ) is a harmonic sphere if for any X ∈ C ∞ (S 2 , R N ) we have Remark 2.2. Harmonic sphere is smooth [H91].
Given a map u : S 2 → M , u ∈ W 1,2 (S 2 , M ), and X ∈ C ∞ (S 2 , R N ), by Taylor polynomial expansion of Π we have the following: By applying ∇ to (4) we have and the energy of u s is HessΠ u (X, ∇u), dΠ u (∇X) HessΠ u (X, ∇u), HessΠ u (X, ∇u) From (5), we see that the first variation of energy is the last equality follows from [Sim96, 2.12.3], and u is a harmonic sphere if and only if (7) ∆u + A(∇u, ∇u) = 0.
The second variation of energy is: HessΠ u (X, ∇u), dΠ u (∇X) HessΠ u (X, ∇u), HessΠ u (X, ∇u) It's clear that from (8) we have Definition 2.3 (Index Form). The index form of a harmonic sphere u : S 2 → M is defined by Definition 2.4 (Index). The index of a harmonic sphere u : S 2 → M is the maximal dimension of the subspace X of Γ(u −1 T M ) on which the index form is negative definite.

2.2.
Bubble convergence of harmonic sphere. This section is for defining bubble convergence (definition 2.7, definition 2.8) and establishing several properties of it (proposition 2.14). They are used for describing how close two maps are, which is essential for lemma 3.1 and theorem 4.1. The varifold distance used by Colding and Minicozzi [CM08] is not sufficient because it only implies closeness in measure sense on the Grassmannian bundle of the ambient manifold. But if a map γ is close to a finite collection of maps {u i } n i=0 in bubble tree sense, that means for each u i there exist conformal dilation D i and compact domain Ω i such that γ is close to u i • D i on Ω i in W 1,2 sense. We state this closeness of bubble convergence in definition 2.8, prove that it implies varifold convergence. Moreover, if a map γ is close to {u i } n i=0 in bubble tree sense if and only if it's close in varifold sense and the term inf is small (see proposition 2.14), this result is used in theorem 4.1.
Definition 2.6 (Möbius transformations). The group of automorphisms of the Riemann sphere is known as P SL(2, C), it's also known as the group of Möbius transformations. Its elements are fractional linear transformations Definition 2.7 (Bubble Convergence). We will say that a sequence γ j : S 2 → M of W 1,2 maps bubble converges to a collection of W 1,2 maps u 0 , ..., u m : S 2 → M if the following hold: (1) The γ j converges weakly to u 0 and there's a finite set S 0 = {x 1 0 , ..., x k 0 0 } ⊂ S 2 so that the γ j converge strongly to u 0 in W 1,2 (K) for any compact set K ⊂ S 2 \ S 0 .
(2) For each i > 0, we get a point x l i ∈ S 0 and a sequence of balls B r i,j (y i,j ) with y i,j → x l i . Further more, let D i,j be the dilation that takes the southern hemisphere to B i,j (y i,j ). Then the map γ j • D i,j converges to u i as in 1.
Definition 2.8. Given a collection of finitely many harmonic spheres {u i } n i=0 , and let E = n i=0 E(u i ). For γ : S 2 → M , we say if we can find conformal dilations D i : S 2 → S 2 , i = 0, ..., n, and pairwise disjoint domains Ω 0 , ..., Ω n , n i=0 Ω i ⊂ S 2 so the following holds: and conformal dilations {D i } n i=0 satisfying (12), (13), and (14). Theorem 2.9 (Bubble convergence for harmonic maps, [Par96]). Let u i : Σ → M be a sequence of harmonic maps from a Riemann surface to a compact Riemannian manifold with bounded energy E 0 . i.e., E(u i ) ≤ E 0 . Then u i bubble converges to a finte collection of harmonic maps {v j } m j=0 Moreover, Actually the sequence doesn't need to be harmonic. It also works for almost harmonic maps like stated in Theorem A.1. Now we introduce varifold distance and state the relation between bubble convergence and varifold convergence. The following definition of varifold distance d V (·, ·) can be found at [CM11, Chapter 3].
Definition 2.10 (Varifold Distance). Fix a closed mainifold M , let be the Grassmannian bundle of (unoriented) k-planes, that is, each fiber P −1 Π (p) is the set of all k-dimensional linear subspaces of the tangent space of M at p. Since G k M is compact, we can choose a countable dense subset h n of all continuous functions on G k M with supremum norm at most one. Given a finite collection of maps .) J f i is the Jacobian of f i , then the varifold distance between them is defined by: Since M is a closed manifold isometrically embedded in R N . We can define varifold distance using R N instead of M . Namely, let be the Grassmanian bundle of k-planes. We can choose a countable dense subset {h n } of all continuous functions on G k R N with supremum norm at most one. If the pairs Then the varifold distance between them is defined by Remark 2.11. We can assume h 0 is constant 1 in definition 2.10. Given two maps u, v : S 2 → M and u, v ∈ W 1,2 (S 2 , M ) ∩ C 0 (S 2 , M ). If d V (u, v) = 0, then it's easy to see by (15), we have Proposition 2.12 (Colding-Minicozzi, [CM08]). If a sequence {γ j } of W 1,2 (S 2 , M ) maps bubble converges to a collection of finitely many smooth maps u 0 , ..., u n : S 2 → M then it also varifold converges to u 0 , ..., u n .
such that the following holds: For each u i , let U i denote the corresponding map to G 2 M . Similarly, for each γ j , let R j denote the corresponding map to G 2 M . We will also use that the map ∇u → J u is continuous as a map from L 2 to L 1 and thus area of u is continuous with respect to energy of u. (see [CM08,proposition A.3]) The proposition now follows by showing for each i and any where the first equality is simply the change of variables formula for integration, and the last equality follows from (18).
Given a collection of harmonic spheres {u i } n i=0 . Since u i : S 2 → M , u i ∈ W 1,2 (S 2 , M ) for each i, and M is a closed Riemannian manifold isometrically embedded in R N , we have that Proof. It follows from for any {φ i } n i=0 ∈ P SL(2, C) and pairewise disjoint domains {Ω i } n i=0 ⊂ S 2 we have the following inequality: Which implies the desired result: Here C(n) = 5 + 2(n + 1)E. Now we know that by Theorem 2.12 bubble convergence implies varifold convergence, but the converse is not true, since varifold convergence simply implies that the images of two maps is close in measure sense. The following Proposition states that varifold convergence implies the term d B (·, ·) goes to zero if the term: Proposition 2.14. Given a collection of finitely many harmonic spheres {u i } n i=0 , for all ǫ > 0, there exists δ > 0 so that if γ ∈ W 1,2 (S 2 , M ) satisfies the following conditions: contains only one harmonic sphere say u 0 , then given ǫ > 0 and let δ = ǫ/2. The condition For the general case it suffices to consider that {u i } n i=0 = {u 0 , u 1 }. We argue by contradiction, suppose there exists ǫ > 0 and a sequence {γ (see definition 2.8). We first consider the case of (20) assuming the following condition i=0,1 We consider the following inequality (see remark 2.11.) For j sufficiently large, we have that which implies that contradicting our assumption (20). Now we consider the case of (21). By our assumption of the sequence {γ j } j∈N : which implies that: Since {φ j 0 } j∈N is diverging, then for j sufficiently large we can find q ∈ S 2 such that (23) |∇u 1 | 2 < ǫ 2 /10, Then the assumption (21) implies From the inequality (24) we have the following: for j sufficiently large. It contradicts with the assumption (23). Thus we have proved proposition 2.14.
Remark 2.15. Combining claim 2.13 and proposition 2.14 we have for all ǫ > 0, there exists δ > 0 such that 3. Statement of Colding and Minicozzi's Min-max thoery. We state some basic notations and min-max theorem in this section.

Definition 2.16 (Width).
Let Ω be the set of continuous maps σ : in a strong sense. Given a map β ∈ Ω, the homotopy class Ω β is defined to be the set of maps σ ∈ Ω that is homotopic to β through maps in Ω. We'll call any such σ a sweepout. The width W = W E (β, M ) associated to the homotopy class Ω β is defined by: We could alternatively define the width using area rather than energy by setting Area(σ(·, t)).
Remark 2.17. We're interested in the case where β induces a map in a nontrivial class in π 3 (M ), in which case the width is positive.
Definition 2.18 (Minimizing sequence). Given a sweepout γ j (·, t) : Definition 2.19. We define the equivalent class of u : S 2 → M to be: Definition 2.21 (Image set). The image set Λ({γ j (·, t)}) of {γ j (·, t)} j∈N is defined to be: Now we state the min-max theorem for harmonic sphere. Theorem 2.22 isn't exactly what's stated in [CM08], it uses d B (·, ·) instead of varifold norm and applies to any minimizing sequence. We prove in appendix A that Colding-Minicozzi's result [CM08] does imply theorem 2.22.
Theorem 2.22 (Min-Max for harmonic sphere). Given a closed manifold M with dimension at least three, and a map β ∈ Ω representing a nontrivial class in π 3 (M ), then for any sequence of sweepouts γ j ∈ Ω β with , and a collection of finitely many harmonic spheres

Unstable Lemma
The main focus of the section is lemma 3.1: proving the energy is concave for maps that are sufficiently close to a finite collection of harmonic spheres in bubble tree sense. We first consider the simplest example, for a given map u ∈ W 1,2 (S 2 , M ), and X ∈ C ∞ (S 2 , R N ) with By the form of second variation of energy (see (8)), clearly if ǫ > 0 is sufficiently small, then for any γ ∈ W 1,2 (S 2 , M ) with γ − u W 1,2 < ǫ, we have Now we consider the general case, given a finite collection of harmonic spheres , positive constant c l > 0 for each l, and the corresponding harmonic spheres v l ∈ {u i } n i=0 , such that for each l. The proof of lemma 3.1 is long and detailed but the idea behind it is simple. It can be roughly spoken as the following: if d B (γ, {u i } n i=0 ) < ǫ for some γ ∈ W 1,2 (S 2 , M ), since v l ∈ {u i } n i=0 , there exist Ω l ⊂ S 2 and D l ∈ P SL(2, C), so that see definition (2.8). If ǫ is sufficiently small the following term is small By choosing a suitable cutoff function η l we can make the following term small LetX l = η l X l • D l and γ s = Π • γ + sX l ), observe that The first term of the right hand side of (27) is close to d 2 ds 2 s=0 S 2 |∇γ s | 2 , and the second term is small because of (26). We have the desired inequality We now state and prove the unstable lemma and specify how to choose ǫ > 0 and η l .
Lemma 3.1 (Unstable lemma). Let M be a closed manifold of dimension at least three, isometrically embedded in R N . Given a collection of finitely many harmonic spheres and let E γ (s) := E(γ s ), so that the following hold: (1) E γ (s) has a unique maximum at m γ ∈ B k c 0 √ 10 (0).
(3) ∀s ∈B k we have Proof. By lemma 3.1, ∀s ∈B k , we have For each s ∈B k , there exists δ(s) > 0 so that for all σ ∈ W 1,2 (S 2 , M ) with we have Proof of the claim. If not, there exists a sequence {s i } i∈N such that lim i→∞ δ(t i ) = 0. SinceB k is compact, we have that lim i→∞ s i = s ′ ∈B k , and δ(s ′ ) > 0 implies the desired contradiction.

Deformation Theorem
Let M be a closed manifold with dimension at least three, isometrically embedded in R N . Consider a map β ∈ Ω representing a nontrivial class in π 3 (M ), let W be the width associated to the homotopy class Ω β (see definition 2.16, (25)), and given a sequence of sweepouts γ j (·, t) ∈ Ω β which is minimizing, i.e., lim j→∞ max t∈[0,1] E(γ j (·, t)) = W.
be a finite set of finite collection of equivalent classes of harmonic spheres, so there exist a constant ǫ k > 0 and j k ∈ N such that d B (γ j (·, t), {k l i } m l i=0 ) > ǫ k , ∀t ∈ [0, 1], for all j > j k , l = 1, ..., N k .
Theorem 4.1 (Deformation Theorem). As assumed above, given a collection of finitely many harmonic spheres {u i } n i=0 with n i=0 Index(u i ) = k > 1 and n i=0 E(u i ) = W . There exists a sequence of sweepouts {γ ′ j (·, t)} j∈N such that Proof. Assumption n i=0 Index(u i ) = k implies there are k correspongding vector fields . Let ǫ > 0 be given as lemma 3.1. By proposition 2.14, there exists δ > 0 such that ) < δ, for γ ∈ W 1,2 (S 2 , M ). We consider the following sets: ) ≤ δ/2 . Let U j,δ/2 := I j,δ ∩ I ′ j,δ/2 . Proposition 2.14 implies that d B (γ j (·, t), {u i } n i=0 ) < ǫ, for all t ∈ U j,δ/2 . We define For t m ∈ U j,δ/2 , by lemma 3.1, we can construct vector fields {X l (t m )} k l=1 , and the hes- here c 0 is a constant given by lemma 3.1. By corollary 3.2 and the continuity of γ j (·, t) in W 1,2 (S 2 , M ) with respect to t, we know there exists δ(t m ) > 0 so that for all t ∈ (t m − δ(t m ), t m + δ(t m )) ∩ U j,δ/2 we have Let I tm := (t m − δ(t m ), t m + δ(t m )) ∩ U j,δ/2 , since U j,δ/2 is compact we can cover U j,δ/2 by finitely many I t , say I t 1 , ..., I t N 1 . Moreover, after discarding some of the intervals, we can arrange that each t is in at least one closed intervalĪ tm , eachĪ tm intersects at most two otherĪ t k 's, and theĪ t k 's intersectingĪ tm do not intersect each other. For each m = 1, ..., N 1 , choose a smooth function ξ m (t) : [0, 1] → [0, 1] which is supported inĪ tm , and We define X l (t) to be The last inequality follows from δ 2 a (t) + δ 2 b (t) ≥ 1/2 By (52), we can choose c = c 0 /2 such that − 1 c Id < D 2 and let E t j (s) := E(γ j,s (·, t)), then we have (1) E t j (s) has a unique maximum at m j (t) ∈ B k c √ 10 (0).
(3) ∀s ∈B k and ∀t ∈ U j,δ/2 we have ) > ǫ l for l = 1, ..., N k , ∀t ∈ [0, 1]. for all j > j k . Without loss of generality (by rescaling {X l (t)} k l=1 and c), we can assume that there exists j ′ k ∈ N such that (55) We argue by contradiction. Assume that there are sequences s ′ i ∈B k and t ′ i ∈ I satisfying For nontrivial harmonic sphere u i : S 2 → M , S 2 \Ω i |∇(u i • D i )| 2 = 0 is only possible when Ω i = S 2 . That implies {u i } n i=0 only contains one harmonic sphere, say u 0 . The index assumption n i=0 Index(u i ) = k now becomes Index(u 0 ) = k, and δ 2 E(u 0 )(X ′ l , X ′ l ) < 0 for l = 1, ..., k. Moreover, the X l (t) we constructed in (51) is simply X ′ l . By γ j,s (·, t) ∈ C 0 (S 2 , M ) ∩ W 1,2 (S 2 , M ), we have the equality holds for all x ∈ S 2 . Moreover, because Π is the nearest point projection defined on a tubular neighborhood of M in R N , which implies the following So we have with the equality holds if and only if s = 0. On the other hand, by assumption (56) which forces s to be 0. Thus γ j (·, t) = u 0 , which contradicts that t belongs to the closure of U j,δ/2 \ U j,δ/3 . Now we consider the one-parameter flow , κ, ǫ so that for any t ∈ U j,δ/2 , and v ∈B k with |v − m j (t)| ≥ κ we have: Proof. By m j (t) ∈ B k c √ 10 (0) and (54) we know that for γ j (·, t), t ∈ U j,δ/2 , we have: So, to prove (60), it suffices to show the existence of T j such that We argue by contradiction and assume that there exists a constant 1 4 > κ > 0, a sequence {t l } l∈N ⊂ U j,δ/2 , and {s l } l∈N ⊂B k with |s l − m j (t l )| ≥ κ such that Combining (62) with (61) we have E t l j (φ t l j (s l , l)) ≥ E t l j (m j (t l )) − c 5 . Since φ t j (·, ·) is an energy decreasing flow, we have Since both U j,δ/2 andB k are compact, we obtain subsequential limits t ∈ U j,δ/2 and s ∈B k with Thus we have lim x→∞ |φ t j (s, x)| = 1 and thus we deduce from the equation (63) that On the other hand, m j (t) ∈ B k c √ 10 (0) implies |v − m j (t)| > 2/3 for all v ∈B k with |v| = 1. Hence, by equation 54 we have which gives us the desired contradiction.
We define a continuous homotopy: We are able to define H ′ j due to the assumption i Index(u i ) = k ≥ 2. So we can choose a continuous path in B k 1/2 j (0) away from the curve of m j (t), t ∈ U j,δ/2 . By claim 4.3, there exists T j for t ∈ U j,δ/2 such that: Let c j : [0, 1] −→ [0, 1] be a cutoff function which is supported in U j,δ/2 , and has value one in U j,δ/3 , value zero in [0, 1] \ U j,δ/2 . Define: We now set s j (t) = (s 1 j (t), ..., s k j (t)) ∈B k to be s j (t) = φ t j (H j (t, 1), c j (t)T j ), if t ∈ U j,δ/2 , and s j (t) = 0, if t ∈ [0, 1] \ U j,δ/2 . We define γ ′ j (·, t) to be: Since s j is homotopic to the zero map inB k , so γ ′ j (·, t) is homotopic to γ j (·, t).
Proof. The claim follows from the assumption 55.
Claim 4.6. there exists ǫ J > 0 and J ∈ N such that Proof. There are three cases to consider.
Theorem 1.1. Let (M, g) be a closed Riemannian manifold of dimension at least three, g generic and a nontrivial homotopy group π 3 (M ), let W be the width associated to the homotopy class Ω β (see definition 2.16, (25)). Then there exists a collection of finitely many harmonic spheres {u i } m i=0 , u i : S 2 → M , which satisfies the following properties: Index(u l i ) > 1 for each l ∈ N. Given a minimizing sequence {γ j (·, t)} j∈N , we consider the collection of harmonic spheres: {[u 1 i ]} n 1 i=0 , and by Theorem 4.1 there exists {γ 1 j (·, t)} j∈N so that (1) γ 1 j (·, t) is homotopic to γ j (·, t), (2) {γ 1 j (·, t)} j∈N is a minimizing sequence, (3) there exists ǫ 1 > 0 and i 1 ∈ N such that We can apply Theorem 4.1 again at the minimizing sequence the given collection of harmonic spheres, and set the compact set of harmonic spheres K to be K 1 := {[u 1 i ] n 1 i=0 } , and obtain {γ 2 j (·, t)} j∈N so that (1) γ 2 j (·, t) is homotopic to γ j (·, t), (2) {γ 2 j (·, t)} j∈N is a minimizing sequence, (3) there exist ǫ 1 , ǫ 2 > 0 and i 1 , i 2 ∈ N such that there exist ǫ l > 0 and i l ∈ N, l = 1, ..., m, such that The goal of this section is to prove theorem 2.22, that any minimizing sequence has a min-max sequence that bubble converges to a finite collection of harmonic spheres. Theorem 2.22 doesn't follow immediately from Colding-Minicozzi's result [CM08, theorem 1.8], which states that there exists a sweepout, so that whenever the area of a slice of the sweepout is close to the width it must be close to a finite collection of harmonic spheres in bubble tree sense itself. [CM08, theorem 1.8] is proven by showing that given any minimizing sequence {γ j (·, t)} j∈N , we can apply harmonic replacement, so that the pulled-tight sequenceγ j (·, t) contains a min-max sequence, which is almost harmonic (see theorem A.1), thus bubble converges to a collection of finitely many harmonic spheres {u i } n i=0 (whose sum of the energies realizes the width). That is, . Sinceγ j (·, t) is obtained from γ j (·, t) by doing harmonic replacement on the disjoint closed balls on S 2 with energy at most ǫ 1 > 0 (ǫ 1 as given in [CM08,theorem 3.1], so by [CM08, theorem 3.1] we have Combining with the assumption γ j is a minimizing sequence, we can conclude that . We first list several technical results in [CM08] and then prove theorem 2.22. Theorem A.1 (Compactness for almost harmonic maps, [CM08]). Suppose that 0 < ǫ ≤ ǫ su (ǫ su is a constant given by [SU81, theorem 3.2]), E 0 > 0 are constants and γ j : there is an energy minimizing map v : B → M that equals to γ j on 1 8 ∂B with Then a subsequence of γ j bubble converges to a finite collection of harmonic spheres u 0 , ..., u m : S 2 → M.
Proposition A.4 (Proposition 1.2 [CM08]). Given a closed manifold M with dimension n ≥ 3, and a map β ∈ Ω representing a nontrivial class in π 3 (M ). The width of energy W , and the width of area W A associated to the homotopy class Ω β are equal.
Remark A.5. Let {γ j } j∈N be a minimizing sequence for W . Since Area(γ j (·, t)) ≤ E(γ j (·, t)), and W A = W by Proposition A.4, we have that which implies that {γ j } j∈N is also a minimizing sequence for W A .
With the above observation, we now state and prove theorem 2.22: Theorem A.8. Given a closed manifold M with dimension at least three, and a map β ∈ Ω representing a nontrivial class in π 3 (M ), then for any sequence of sweepouts there exists a subsequence {i j } → ∞, t i j ∈ [0, 1], and a collection of finitely many harmonic spheres E(γ j (·, s)) < W + 1/2j.
We are going to prove in this section that for a closed manifold of dimension at least three, with a generic metric, then the set of all harmonic spheres up to equivalent class with bounded energy is countable (proposition B.25). This result is expected, given that we know a similar result holds for minimal embedded hypersurfaces. Namely: . Given a closed manifold (M n , g), 3 ≤ n ≤ 7, with g generic, the set of embedded minimal hypersurfaces with bounded area and index is finite.
However, harmonic spheres with bounded energy are merely branched minimal immersions with bounded area. Proposition B.25 doesn't follow from theorem B.1. The proof of Proposition B.25 brings together several important results like bumpy metric theorem for minimal surface [Whi17, theorem 2.1], bumpy metric theorem for harmonic map [Moo17] and comparison between second variation of area and energy of a minimal surface [EM08]. We first recall some basic properties for harmonic maps like tension field and Jacobi field, state bumpy metric theorem for harmonic maps and minimal submanifold, and then prove proposition B.25. Remark B.3. Once we know that ∇u ∈ L ∞ loc , it then follows from equation (7) that ∆u ∈ L ∞ loc , which implies by standard estimates on the inverse of the Laplacian that u ∈ W 2,p loc for all p < ∞. Hence we deduce that ∆u ∈ W 1,p loc and hence u ∈ W 3,p loc for all p > 0. We can then repeat this argument to show that u ∈ W r,p loc , ∀r, and so the smoothness of the solution follows.
here ∆ is the Laplacian on sections of f −1 T N given in local coordinates on M by be a smooth family of maps between Riemannian manifolds of finite energy. M may have nonempty boundary, in which case we require f (x, s, t) = f (x, 0, 0) for all x ∈ ∂M and all s, t.
Definition B.10 (Bumpy Metric [Whi17]). A metric g on M is called bumpy if there is no smooth immersed minimal submanifold (minimal with respect to g) with a non-trivial Jacobi field.
Theorem B.11 (Bumpy Metric Theorem for Minimal Submanifold, [Whi17]). If M is a compact manifold, then for a generic choice of metric of C q g on M (q ≥ 3), there are no minimal submanifolds with nonzero normal Jacobi fields. That is, each minimal submanifold has nullity 0.  Definition B.14. If p is a branch point of f but there exists some neighborhood V containing p such that f (V ) is an immersed surface, then we say that p is a false branch point.
If f : Σ → M is connected and has injective points, we say it is somewhere injective. Remark B.19. In Theorem B.18, nondegeneracy of a prime harmonic map means that the Jacobi field of it are those generated by the automorphisms of Σ.
In our case Σ = S 2 and G = P SL(2, C). Given a harmonic sphere u : S 2 → M , if u is prime then it's free of branch points by Theorem B.18. If u is not prime it can be written as a branched cover of a prime harmonic sphere, and all of its branched points are false. Theorem B.18 implies that for all harmonic spheres u : S 2 → M , u(S 2 ) is an smooth immersed minimal submanifold.
We define the nullity N of a functional at a critical point u is the dimension of the space of Jacobi fields of the functional at u. Proof. Given a harmonic sphere u : S 2 → M , Theorem B.18 implies that u(S 2 ) is a smooth immersed minimal sphere. We argue by contradiction. If not, then there exists a sequence of harmonic spheres {f i } i∈N , f i : S 2 → M , such that f i − u W 1,2 < 1/i and [f i ] = [u] for all i ∈ N. By strong convergence in W 1,2 , Theorem B.2, and Arzelà-Ascoli theorem we know that the convergence f i → u is smooth and uniform. Thus we can choose subsequence i(j) → ∞ as j → ∞ such that (72) |f i(j) (x) − u(x)| < 1/j, ∀x ∈ S 2 , and (73) ∂f i(j) (x) ∂x α − ∂u(x) ∂x α < 1/j, ∀x ∈ S 2 , where (x α ) denotes the local coordinate system of S 2 . Since f i(j) is harmonic we have that Thus there exists t j ∈ (0, 1) such that 0 = ∂ ∂t t=t j τ (u + t(f i(j) − u)).
By proposition B.7, we know that w is an nontrivial Jacobi field. i.e., J u (w) = 0. Assume that f i(j) (S 2 ) = u(S 2 ) for all j ∈ N. Since f i (S 2 ) = u(S 2 ) implies that w is not purely tangential, Theorem B.20 implies that u has a nontrivial Jacobi field for area functional as a smooth immersed minimal surface. This contradicts the bumpy metric assumption. Now we consider the case that f i(j) (S 2 ) = u(S 2 ) for all j ∈ N. If u is a prime harmonic sphere, then Theorem B.18 implies that w is generated by P SL(2, C), contradicting the assumption [f i(j) ] = [u]. Now we consider the case that u is not prime. Proof of the claim. It's known that if two maps are homotopic then they have the same degree. The stradegy of the proof is similar and can be found in [Hir94,Chapter 5].
By the smooth convergence of f i(j) → u, and deg(f i(j) ) = deg(u). Let u = φ •ũ for some prime harmonic mapũ. We can write f i j as a branched cover of a prime harmonic map h i j ,i.e., f i j = g i j • h i j , with deg(f i(j) ) = deg(g i j ) = deg(u). By lemma B.16 and lemma B.17 we know that h i j is somewhere injective and the injective points of h i j form an open dense subset of S 2 . So we obtain a sequence of prime harmonic spheres h i j that converges strongly to a prime harmonic sphereũ in W 1,2 . It is the desired contradiction.  Definition B.24. We define F W to be the equivalent classes of all harmonic sphere with energy bound W , that is: Proposition B.25. Given a closed manifold (M, g), with generic g and the dimension of M is at least three. The set F W is countable.
Proof. We pick a finite set {x 1 n , ..., x p n n } of S 2 so that S 2 ⊂ p n k=1 B 1/n (x k n ). (B 1/n (x k n ) is the geodesic ball centered at the point x k n .) We define F W (n) to be: |∇f | 2 < ǫ su , for k = 1, ..., p n , where ǫ su > 0 is the constant given in Theorem B.2. We can see that: |∇f i | 2 < ǫ su ∀i ∈ N, k = 1, ..., p n .
By Theorem B.2, this implies that the convergence is strong in W 1,2 and f ∈ F W (n).
Then it contradicts Lemma B.21.