1 Introduction

Finite-dimensional Morse theory was developed by Morse [10] to study geodesics. Index of a critical points 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 \(\alpha \)-energy [19], which can be perturbed to be Morse functions. The \(\alpha \)-energy approaches the usual energy as the parameter \(\alpha \) in the perturbation goes to one, and the corresponding critical points of \(\alpha \)-energy converges to a harmonic map. However, without curvature assumption [9] or finite fundamental group [4] for the ambient manifold (M, g), the harmonic spheres constructed by \(\alpha \)-energy fails to realize the energy as \(\alpha \) goes to one [8] [13, Remark 4.9.6]. Thus, we are motivated to prove the Morse index bound of the harmonic spheres produced by the min–max theory [1], 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 \(\pi _3(M).\) Then there exists a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n,\, u_i:S^2\rightarrow M\), which satisfies the following properties:

  1. (1)

    \(\sum _{i=0}^n E(u_i)=W,\)

  2. (2)

    \(\sum _{i=0}^n Index(u_i)\le 1,\)

here W is a geometric invariant called width. (See Definition 2.15.)

The collection of finitely many harmonic spheres in Theorem 1.1 is constructed by Colding and Minicozzi [1] 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 follows: given a closed manifold M, M is swept out by a continuous one parameter family of maps from \(S^2\) to M, starting and ending at point maps. The sweepout (Definition 2.15) is pulled tight, in a continuous way, by harmonic replacements. Then if the sweepout induces a non-trivial class in \(\pi _3(M)\), 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 2.10). 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’ min–max theory [15] 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 [11]. This result plays an important role in proving Yau’s conjecture [18], 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 [1], 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 [1].

Theorem 1.1 seems like a variant of [11]. 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 [11] is given by Almgren-Pitts’ min–max theory; thus, it 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 [11] 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 [12]:

Morse Index Conjecture For generic metric on \(M^{n+1}\), \(3\le (n+1)\le 7\), there exists a smooth, embedded, two-sided and closed minimal hypersurface \(\Sigma \) such that \(Index(\Sigma )=k\) for any integer k.

Marques and Neves have shown that the conjecture is true in [12] under the following assumption: there exists a smooth, embedded, two-sided and closed minimal hypersurface which realizes the min–max width. The above assumption is known as Multiplicity One Conjecture and is proved by Zhou in [22].

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 2.20) is one harmonic sphere. In other words, if the min–max sequence (see Definition 2.17) 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 that lie in the small tubular neighborhood. But bubble convergence doesn’t imply the min–max that 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 \(\Lambda (\{\gamma _j(\cdot ,t)\}_j)\) of a minimizing sequence \(\{\gamma _j(\cdot ,t)\}_{j\in {\mathbb {N}}}\). The idea is if \(\{u_i\}_{i=0}^m\in \Lambda (\{\gamma _j(\cdot .t)\}_j)\) have \(\sum _{i=0}^m\text {Index}(u_i)>1\), then we are able to perturb \(\{\gamma _j(\cdot ,t)\}_{j}\) to a new sweepout \(\{{\tilde{\gamma }}_j(\cdot ,t)\}_j\) such that it is homotopic to \(\gamma _j\), it is a minimizing sequence, and \(\{u_i\}_{i=0}^m\) is not in its image set. Since \(\{{\tilde{\gamma }}_j(\cdot ,t)\}_j\) is minimizing, \(\Lambda \{{\tilde{\gamma }}_j(\cdot ,t)\}_j\) is non-empty. If for \(\{v_i\}_{i=0}^{m_0}\in \Lambda (\{{\tilde{\gamma }}_j(\cdot ,t)\}_j)\) we have \(\sum _{i=0}^{m_0}\text {Index}(v_i)>1\), then we can perturb \(\{{\tilde{\gamma }}_j(\cdot ,t)\}_j\) again and get a new sweepout such that neither \(\{u_i\}_{i=0}^m\) nor \(\{v_i\}_{i=0}^{m_0}\) is in its image set. Proposition B.24 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 \(\{{\tilde{\gamma }}_j(\cdot ,t)\}_j\), we define the variation of a map. Suppose M is isometrically embedded in \({\mathbb {R}}^N\) and let \(\Pi :{\mathbb {R}}^N\rightarrow M\) be the nearest point projection from \({\mathbb {R}}^N\) to M. Given a map \(u:S^2\rightarrow M\) and \(X:S^2\rightarrow {\mathbb {R}}^N\), with each \(X^i\in C^{\infty }(S^2)\), we consider the variation of u with respect to X to be \(u_s:=\Pi \circ (u+sX)\). We choose to define the variation this way so that for any map \(v:S^2\rightarrow M\) close to u in \(W^{1,2}(S^2,M)\), the variation \(v_s\) is close to \(u_s\) as well.

Assume \(\{u_i\}_{i=0}^m\in \Lambda (\{\gamma _j(\cdot ,t)\}_j)\) with \(\sum _{i=0}^m\text {Index}(u_i)=k\ge 2\), then there exists \(\{X_l\}_{l=1}^k\), \(X_l:S^2\rightarrow {\mathbb {R}}^N\), with the following property: for each \(X_l\), there exists at least one \(u_l\in \{u_i\}_{i=0}^m\) so the second variation of energy of \(u_l\) with respect to \(X_l\) is negative. The idea is using \(\{X_l\}_{l=1}^k\) to perturb \(\gamma _j(\cdot ,t)\). We first prove in Lemma 3.1 that for \(\gamma _j(\cdot ,t)\) close to \(\{u_i\}_{i=0}^m\), there exist corresponding cutoff functions \(\eta ^j_l:S^2\rightarrow {\mathbb {R}}\). Let \({\tilde{X}}_l:=\eta _l^jX_l\) and define the variation of \(\gamma _j(\cdot ,t)\) with respect to \({\tilde{X}}_l\) to be:

$$\begin{aligned} \gamma _{j,s}(\cdot ,t):=\Pi \circ \big (\gamma _j(\cdot ,t)+\sum _{l=1}^ks_l{\tilde{X}}_l\big ), \end{aligned}$$

here \(s=(s_1,\ldots ,s_k)\in {\bar{B}}^k\), \({\bar{B}}^k\) is the k-dimensional unit ball, so that the energy of \(\gamma _{j,s}(\cdot ,t)\) is concave while changing \(s\in {\bar{B}}^k\). That is, define \(E_j^t(s):=E(\gamma _{j,s}(\cdot ,t)), E_j^t:{\bar{B}}^k\rightarrow {\mathbb {R}},\) and we have

$$\begin{aligned} D^2E_j^t(s)<0,\quad \forall s\in {\bar{B}}^k. \end{aligned}$$

If we can construct a continuous function \(s_j:[0,1]\rightarrow {\bar{B}}^k\) so that energy decreases by a certain amount when \(\gamma _j(\cdot ,t)\) is close to \(\{u_i\}_{i=0}^m\), then the sequence \(\{\gamma _{j,s_j(t)}(\cdot ,t_j)\}_j\) does not converge to \(\{u_i\}_{i=0}^m\), and \(\gamma _{j,s_j(t)}(\cdot ,t)\) is the desired sweepout. In order to construct \(s_j(t)\), we observe the following one parameter gradient flow \(\{\phi _j^t(\cdot ,x)\}\in \text {Diff}({\bar{B}}^k)\), with \(x\in {\bar{B}}^k\) as starting point, generated by the vector field:

$$\begin{aligned} s\mapsto -(1-|s|^2)\nabla E_j^t(s),\quad s\in {\bar{B}}^k. \end{aligned}$$

\(\phi _j^t(\cdot ,x)\) decreases the energy, except when \(|x|=1\) or x is the maximal point of \(E_j^t\). The assumption of the lower bound of Morse index \(\sum _{i=0}^m\text {Index}(u_i)=k>1\), which now enables us to construct a continuous curve \(y_j:[0,1]\rightarrow {\bar{B}}^k\) avoiding the maximal point of the function \(E^t_j\) as t varies. Namely, let \(\nabla E_j^t(x(t))=0\), \(x_j:[0,1]\rightarrow {\bar{B}}^k\), x(t) is a continuous curve on \({\bar{B}}^k\). Since the dimension of \({\bar{B}}^k\) is larger than 1, we can choose a continuous curve \(y_j(t)\) on \({\bar{B}}^k\) which does not intersect with \(x_j([0,1])\), then we can use \(\{\phi _j^t(\cdot ,y_j(t))\}\) to construct \(s_j(t)\) and obtain the new sweepout \({\tilde{\gamma }}_j(\cdot ,t):=\gamma _{j,s_j(t)}(\cdot ,t)\). The new sweepout is homotopic to \(\{\gamma _j(\cdot ,t)\}_j\) and doesn’t bubble converge to \(\{u_i\}_{i=0}^m\). This is the desired perturbed sweepout.

The organization of the paper is as follows. In Sect. 2 we give the basic definitions of harmonic sphere, bubble convergence, and state the min–max Theorem 2.21. In Sect. 3 we prove a technical Lemma 3.1. In Sect. 4 we prove the main result Theorem 1.1.

2 Background Material

2.1 Harmonic Sphere

Suppose that \(S^2\) is a Riemann sphere, which can be regarded as \({\mathbb {C}}\cup \{\infty \}\), and M is a closed manifold of dimension at least three, isometrically embedded in \({\mathbb {R}}^N\).

We introduce nearest point projection \(\Pi :{\mathbb {R}}^N\rightarrow M\) which maps a point \(x\in {\mathbb {R}}^N\) to the nearest point of M. There is a tubular neighborhood of M

$$\begin{aligned} M_\delta =\big \{x\in {\mathbb {R}}^N:\text { dist}(x,M)<\delta \big \}, \end{aligned}$$

on which \(\Pi \) is well-defined and smooth. For a map \(u:S^2\rightarrow M\subseteq {\mathbb {R}}^N\), the energy of u is simply

$$\begin{aligned} E(u)=\int _{S^2}|\nabla u|^2, \end{aligned}$$
(1)

where \(\nabla \) is the gradient on \(S^2\). For a given \(X\in C^{\infty }(S^2,{\mathbb {R}}^N)\), we consider the variation of u with respect to X defined as the following:

$$\begin{aligned} u_{s}=\Pi \circ (u+sX), \end{aligned}$$
(2)

\(u_s\) is well-defined for s small enough such that the image of \(u+sX\) is in the tubular neighborhood \(M_\delta \).

Definition 2.1

(Harmonic Sphere) We say that \(u\in W^{1,2}(S^2,M)\) is a harmonic sphere if for any \(X\in C^{\infty }(S^2,{\mathbb {R}}^N)\) we have

$$\begin{aligned} \lim _{s\rightarrow 0}\frac{E(u_s)-E(u)}{s}=0. \end{aligned}$$
(3)

Remark 2.2

Harmonic sphere is smooth [6].

Given a map \(u:S^2\rightarrow M\), \(u\in W^{1,2}(S^2,M)\), and \(X\in C^{\infty }(S^2,{\mathbb {R}}^N)\), by Taylor polynomial expansion we have the following:

$$\begin{aligned} \begin{aligned} u_s&=\Pi \circ (u+sX)\\&=u+s\mathrm{{d}}\Pi _u(X)+\frac{s^2}{2}\text {Hess}\Pi _u(X,X)+o(s^3). \end{aligned} \end{aligned}$$
(4)

By applying \(\nabla \) to (4), we have

$$\begin{aligned} \nabla u_s&=\nabla u\\&\quad +\,s(\mathrm{{d}}\Pi _u(\nabla X)+\text {Hess}\Pi _u(X,\nabla u))\\&\quad +\,\frac{s^2}{2}\big (2\text {Hess}\Pi _u(X,\nabla X)+\nabla \text {Hess}\Pi _u(X,X,\nabla u)\big )+o(s^3), \end{aligned}$$

and the energy of \(u_s\) is

$$\begin{aligned} \begin{aligned} E(u_s)&=\int _{S^2}\langle \nabla u,\nabla u\rangle \\&\quad +\,2s\int _{S^2}\langle \nabla u,\mathrm{{d}}\Pi _u(\nabla X)\rangle +\langle \nabla u, \text {Hess}\Pi _u(X,\nabla u)\rangle \\&\quad +\,s^2\Big \{\int _{S^2}\langle \mathrm{{d}}\Pi _u(\nabla X),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\text {Hess}\Pi _u(X,\nabla u)\rangle \\&\quad +\,2\int _{S^2}\langle \nabla u,\big ( 2\text {Hess}\Pi _u(X,\nabla X)+\nabla \text {Hess}\Pi _u(X,X,\nabla u)\big )\rangle \Big \}\\&\quad +\,o(s^3), \end{aligned} \end{aligned}$$
(5)

and we have

$$\begin{aligned} \begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}s} E(u_s)&= 2\int _{S^2}\langle \nabla u,\mathrm{{d}}\Pi _u(\nabla X)\rangle +\langle \nabla u, \text {Hess}\Pi _u(X,\nabla u)\rangle \\&\quad +\,2s\Big \{\int _{S^2}\langle \mathrm{{d}}\Pi _u(\nabla X),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\text {Hess}\Pi _u(X,\nabla u)\rangle \\&\quad +\,2\int _{S^2}\langle \nabla u,\big ( 2\text {Hess}\Pi _u(X,\nabla X)+\nabla \text {Hess}\Pi _u(X,X,\nabla u)\big )\rangle \Big \}\\&\quad +\,o(s^2), \end{aligned} \end{aligned}$$
(6)

and

$$\begin{aligned} \begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2} E(u_s)&= 2\Big \{\int _{S^2}\langle \mathrm{{d}}\Pi _u(\nabla X),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\mathrm{{d}}\Pi _u(\nabla X)\rangle \\&\quad +\,\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\text {Hess}\Pi _u(X,\nabla u)\rangle \\&\quad +\,2\int _{S^2}\langle \nabla u,\big ( 2\text {Hess}\Pi _u(X,\nabla X)+\nabla \text {Hess}\Pi _u(X,X,\nabla u)\big )\rangle \Big \}\\&\quad +\,o(s). \end{aligned} \end{aligned}$$
(7)

From (6), we can see that the first variation of energy is

$$\begin{aligned} \begin{aligned} \delta E(u)(X)&:=\frac{\mathrm{{d}}}{\mathrm{{d}}s}\Big |_{s=0}E(u_s)\\&=\int _{S^2}\langle \nabla u,\mathrm{{d}}\Pi _u(\nabla X)\rangle +\langle \nabla u, \text {Hess}\Pi _u(X,\nabla u)\rangle \\&=\int _{S^2}\langle \nabla u, \nabla X\rangle -\langle X,A(\nabla u,\nabla u)\rangle . \end{aligned} \end{aligned}$$
(8)

the last equality follows from [17, 2.12.3]; thus, the map u is a harmonic sphere if and only if

$$\begin{aligned} \Delta u+ A(\nabla u,\nabla u)=0. \end{aligned}$$
(9)

The second variation of energy follows from (7):

$$\begin{aligned} \delta ^2 E(u)(X,X)&:=\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}E(u_s)\nonumber \\&=2\int _{S^2}\langle \mathrm{{d}}\Pi _u(\nabla X),\mathrm{{d}}\Pi _u(\nabla X)\rangle \nonumber \\&\quad +\,4\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\mathrm{{d}}\Pi _u(\nabla X)\rangle \nonumber \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _u(X,\nabla u),\text {Hess}\Pi _u(X,\nabla u)\rangle \nonumber \\&\quad +\,2\int _{S^2}\langle \nabla u,\big ( 2\text {Hess}\Pi _u(X,\nabla X)+\nabla \text {Hess}\Pi _u(X,X,\nabla u)\big )\rangle . \end{aligned}$$
(10)

It’s clear that from (10) we have

$$\begin{aligned} |\delta ^2E(u)(X,X)-\delta ^2E(v)(X,X)|<\Psi (\Vert u-v\Vert _{W^{1,2}}), \end{aligned}$$
(11)

for some continuous function \(\Psi :[0,\infty )\rightarrow [0,\infty )\) which depends on \(\{u,X\}\), with \(\Psi (0)=0\). If u is a harmonic sphere, we have

$$\begin{aligned} \begin{aligned} \delta ^2E(u)(X,X)&=\int _{S^2}\langle \nabla \mathrm{{d}}\Pi _u(X),\nabla \mathrm{{d}}\Pi _u(X)\rangle \\&\quad -\,\int _{S^2} \langle R^M(\nabla u,\mathrm{{d}}\Pi _u(X))\mathrm{{d}}\Pi _u(X),\nabla u\rangle . \end{aligned} \end{aligned}$$
(12)

Definition 2.3

(Index Form) The index form of a harmonic sphere \(u:S^2\rightarrow M\) is defined by

$$\begin{aligned} \begin{aligned} I(X,Y)&=\int _{S^2}\langle \nabla \mathrm{{d}}\Pi _u(X),\nabla \mathrm{{d}}\Pi _u(Y)\rangle \\&\quad -\,\int _{S^2} \langle R^M(\nabla u,\mathrm{{d}}\Pi _u(X))\mathrm{{d}}\Pi _u(Y),\nabla u\rangle , \end{aligned} \end{aligned}$$
(13)

for \(X,Y\in C^\infty (S^2,{\mathbb {R}}^N)\)

Definition 2.4

(Index) The index of a harmonic sphere \(u:S^2\rightarrow M\) is the maximal dimension of the subspace X of \(\Gamma (u^{-1}TM)\) on which the index form is negative definite.

Remark 2.5

([17]) For any \(X\in C^\infty (S^2,{\mathbb {R}}^N)\),

$$\begin{aligned} \mathrm{{d}}\Pi _u(X)\in \Gamma (u^{-1}TM). \end{aligned}$$

2.2 Bubble convergence of Harmonic Sphere

This section is for defining bubble convergence (Definitions 2.7 and 2.8) and establishing several properties of it. 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 [1] is not sufficient because it only implies closeness in measure sense on the Grassmannian bundle of the ambient manifold. But if a map \(\gamma \) is close to a finite collection of maps \(\{u_i\}_{i=0}^n\) in bubble tree sense, that means for each \(u_i\) there exist conformal dilation \(D_i\) and compact domain \(\Omega _i\) such that \(\gamma \) is close to \(u_i\circ D_i\) on \(\Omega _i\) in \(W^{1,2}\) sense. We state this closeness of bubble convergence in Definition 2.8 and prove that it implies varifold convergence.

Definition 2.6

(Möbius transformations) The group of automorphisms of the Riemann sphere is known as \(PSL(2,{\mathbb {C}})\), it’s also known as the group of Möbius transformations. Its elements are fractional linear transformations

$$\begin{aligned} \phi (z)=\frac{az+b}{cz+d},\quad ad-bc\ne 0, \end{aligned}$$

where \(a,b,c,d\in {\mathbb {C}}.\)

Definition 2.7

(Bubble Convergence) We will say that a sequence \(\gamma _{j}:S^2\rightarrow M\) of \(W^{1,2}\) maps bubble converges to a collection of \(W^{1,2}\) maps \(u_0,\ldots ,u_m:S^2\rightarrow M\) if the following hold:

  1. (1)

    The \(\gamma _{j}\) converges weakly to \(u_0\) and there’s a finite set \({\mathcal {S}}_0=\{x_0^1,\ldots ,x_0^{k_0}\}\subset S^2\) so that the \(\gamma _{j}\) converge strongly to \(u_0\) in \(W^{1,2}(K)\) for any compact set \(K\subset S^2\setminus {\mathcal {S}}_0\).

  2. (2)

    For each \(i>0\), we get a point \(x_{l_i}\in {\mathcal {S}}_0\) and a sequence of balls \(B_{r_{i,j}}(y_{i,j})\) with \(y_{i,j}\rightarrow x_{l_i}\). Furthermore, let \(D_{i,j}\) be the dilation that takes the southern hemisphere to \(B_{r_{i,j}}(y_{i,j})\). Then the map \(\gamma _{j}\circ D_{i,j}\) converges to \(u_i\) as in 1.

  3. (3)

    if \(i_1\ne i_2\), then \(\frac{r_{i_1,j}}{r_{i_2,j}}+\frac{r_{i_2,j}}{r_{i_1,j}}+\frac{|y_{i_1,j}-y_{i_2,j}|^2}{r_{i_1,j}r_{i_2,j}}\rightarrow \infty .\)

  4. (4)

    \(\sum _{i=0}^m E(u_i)=\lim \limits _{j\rightarrow \infty }E(\gamma _{j}).\)

We introduce \(d_B(\cdot ,\cdot )\) here to describe bubble convergence precisely. Notice that \(d_B(\cdot ,\cdot )\) is not a norm like \(\Vert \cdot \Vert _{W^{1,2}}\) or \(d_V(\cdot ,\cdot )\) (see Definition 2.11), we are abusing the notation here by using \(d_B(\cdot ,\cdot )\).

Definition 2.8

Given a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\), and let \(E=\sum _{i=0}^nE(u_i)\). For \(\gamma :S^2\rightarrow M\), we say

$$\begin{aligned} d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon , \end{aligned}$$

if we can find conformal dilations \(D_{i}:S^2\rightarrow S^2,\,i=0,\ldots ,n,\) and pairwise disjoint domains \(\Omega _{0},\ldots ,\Omega _{n}\), \(\bigcup _{i=0}^n\Omega _{i}\subset S^2\) so the following holds:

$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{\Omega _{i}}|\nabla \gamma -\nabla (u_i\circ D_{i})|^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(14)
$$\begin{aligned}&\Big (\int _{S^2\setminus \bigcup _{i=0}^n\Omega _{i}}|\nabla \gamma |^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(15)
$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{S^2\setminus \Omega _{i}}|\nabla (u_i\circ D_{i})|^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(16)

We write \(d_B(\gamma ,\{u_i\}_{i=0}^n)\ge \epsilon \) if there’s no pairwise disjoint domains \(\{\Omega _i\}_{i=0}^n\) and conformal dilations \(\{D_i\}_{i=0}^n\) satisfying (14), (15), and (16).

Remark 2.9

If \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \), then we have

$$\begin{aligned} \begin{aligned} \int _{S^2}|\nabla \gamma |^2&=\sum _{i=0}^n\int _{\Omega _i}|\nabla (u_i\circ D_{i})-\big (\nabla \gamma -\nabla (u_i\circ D_{i})\big )|^2\\&\quad +\,\int _{S^2\setminus \bigcup _{i=0}^n\Omega _{i}}|\nabla \gamma |^2\\&\le \sum _{i=0}^n\Big \{\int _{\Omega _i}|\nabla (u_i\circ D_{i})|^2+2\int _{\Omega _i}|\nabla (u_i\circ D_{i})||\nabla \gamma -\nabla (u_i\circ D_{i})|\\&\quad +\,\int _{\Omega _i}|\nabla \gamma -\nabla (u_i\circ D_{i})|^2\Big \}+\int _{S^2\setminus \bigcup _{i=0}^n\Omega _{i}}|\nabla \gamma |^2\\&\le \sum _{i=0}^nE(u_i)+2\big (\sum _{i=0}^nE(u_i)\big )^{1/2}\epsilon +\epsilon ^2. \end{aligned}\nonumber \\ \end{aligned}$$
(17)

which implies that

$$\begin{aligned} \left| \sum _{i=0}^n E(u_i)-E(\gamma )\right| \le 2\left( \sum _{i=0}^nE(u_i)\right) ^{1/2}\epsilon +\epsilon ^2. \end{aligned}$$
(18)

Theorem 2.10

(Bubble convergence for harmonic maps [14]) Let \(u_i:\Sigma \rightarrow M\) be a sequence of harmonic maps from a Riemann surface to a compact Riemannian manifold with bounded energy \(E_0\). i.e.,

$$\begin{aligned} E(u_i)\le E_0. \end{aligned}$$

Then \(u_i\) bubble converges to a finte collection of harmonic maps \(\{v_j\}_{j=0}^m\) Moreover,

$$\begin{aligned} \lim \limits _{i \rightarrow \infty }E(u_i)=\sum _{j=0}^mE(v_j). \end{aligned}$$

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 \(d_V(\cdot ,\cdot )\) can be found at [2, Chapter 3].

Definition 2.11

(Varifold Distance) Fix a closed mainfold M, let

$$\begin{aligned} P_\Pi :{\mathcal {G}}_kM\rightarrow M \end{aligned}$$

be the Grassmannian bundle of (unoriented) k-planes, that is, each fiber \(P_\Pi ^{-1}(p)\) is the set of all k-dimensional linear subspaces of the tangent space of M at p. Since \({\mathcal {G}}_kM\) is compact, we can choose a countable dense subset \({h_n}\) of all continuous functions on \({\mathcal {G}}_kM\) with supremum norm at most one. Given a finite collection of maps

$$\begin{aligned} f_i&:X_i\rightarrow M,\quad f_i\in W^{1,2}(X_i,M),i=1,\ldots ,k_1,\\ g_j&:Y_j\rightarrow M,\quad g_j\in W^{1,2}(Y_j,M),j=1,\ldots ,k_2, \end{aligned}$$

here \(\{X_i\}_{i=1}^{k_1},\{Y_j\}_{j=1}^{k_2}\) are compact surfaces of dimension k. We consider the pairs \(\{(X_i,F_i)\}_{i=1}^{k_1}\) and \(\{(Y_j,G_j)\}_{j=1}^{k_2}\) with measurable maps

$$\begin{aligned} F_i:X_i\rightarrow {\mathcal {G}}_kM,\quad \text {and }G_j:Y_i\rightarrow {\mathcal {G}}_kM, \end{aligned}$$

so that

$$\begin{aligned} f_i=P_\Pi \circ F_i,\quad \text {and }g_j=P_\Pi \circ G_j. \end{aligned}$$

(\(F_i(x)\) is the linear subspace of \(df_i(T_xM)\).) \(J_{f_i}\) is the Jacobian of \(f_i\), then the varifold distance between them is defined as follows:

$$\begin{aligned} d_V(\{f_i\}_{i=1}^{k_1},\{g_j\}_{j=1}^{k_2}):=\sum _{n=0}^{\infty }2^{-n}\Big |\sum _{i=1}^{k_1}\int _{X_i}h_n\circ F_iJ_{f_i}-\sum _{j=1}^{k_2}\int _{Y_j}h_n\circ G_jJ_{g_j}\Big |.\qquad \end{aligned}$$
(19)

Remark 2.12

We can assume \(h_0\) is constant 1 in Definition 2.11. Given two maps \(u,v:S^2\rightarrow M\) and \(u,v\in W^{1,2}(S^2,M)\cap C^0(S^2,M)\). If \(d_{V}(u,v)=0\), then it’s easy to see by (19) that we have \({Area}(u)=\text {Area}(v)\).

Proposition 2.13

(Colding–Minicozzi [1]) If a sequence \(\{\gamma _j\}\) of \(W^{1,2}(S^2,M)\) maps bubble converges to a collection of finitely many smooth maps \(u_0,\ldots ,u_n:S^2\rightarrow M\) then it also varifold converges to \(u_0,\ldots ,u_n\).

Remark 2.14

Proposition 2.13 is proved in [1]. We prove it again using the notation \(d_B(\cdot ,\cdot )\) we introduced instead for the completeness.

Proof

Let \(E=\sum _{i=0}^nE(u_i)\), since \(\gamma _j\) bubble converges to \(\{u_i\}_{i=0}^n\), we assume without loss of generality that \(d_B(\gamma _j,\{u_i\}_{i=0}^n)<1/j\). So there exists conformal dilations \(D^j_{i}\in PSL(2,{\mathbb {C}}),\,i=0,\ldots ,n,\) and pairwise disjoint domains \(\Omega ^j_{0},\ldots ,\Omega ^j_{n}\), \(\bigcup _{i=0}^n\Omega ^j_{i}\subset S^2.\) such that the following holds:

$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{\Omega ^j_{i}}|\nabla (\gamma _j-\nabla (u_i\circ D^j_{i})|^2\Big )^{1/2}<1/j, \end{aligned}$$
(20)
$$\begin{aligned}&\int _{S^2\setminus \bigcup _{i=0}^n\Omega ^j_{i}}\Big (|\nabla \gamma _j|^2\Big )^{1/2}<1/j, \end{aligned}$$
(21)
$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{S^2\setminus \Omega ^j_{i}}|\nabla (u_i\circ D^j_{i})|^2\Big )^{1/2}<1/j. \end{aligned}$$
(22)

For each \(u_i\), let \(U_i\) denote the corresponding map to \({\mathcal {G}}_2M\). Similarly, for each \(\gamma _j\), let \(R_j\) denote the corresponding map to \({\mathcal {G}}_2M\). We will also use that the map \(\nabla u\rightarrow J_u\) 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 [1, proposition A.3])

The proposition now follows by showing for each i and any \(h\in C^0({\mathcal {G}}_2M)\) that

$$\begin{aligned} \sum _{i=0}^n\int _{S^2}h\circ U_i J_{u_i}&=\sum _{i=0}^n\lim _{j\rightarrow \infty }\int _{\Omega ^j_{i}}h\circ U_i\circ D^j_i J_{u_i\circ D^j_i}\\&=\sum _{i=0}^n\lim _{j\rightarrow \infty }\int _{\Omega ^j_{i}} h\circ R_j J_{\gamma _j}\\&=\lim _{j\rightarrow \infty }\int _{\cup _i\Omega ^j_{i}}h\circ R_j J_{\gamma _j}\\&=\int _{S^2}h\circ R_j J_{\gamma _j}, \end{aligned}$$

where the first equality is simply the change of variables formula for integration, and the last equality follows from (21). \(\square \)

2.3 Statement of Colding and Minicozzi’s Min–Max Theory

We state some basic notations and min–max theorem in this section.

Definition 2.15

(Width) Let \(\Omega \) be the set of continuous maps \(\sigma :S^2\times [0,1] \rightarrow M\) so that for each \(t\in [0,1]\) the map \(\sigma (\cdot ,t)\) is in \(C^0(S^2,M)\cap W^{1,2}(S^2,M)\), the map \(t\rightarrow \sigma (\cdot ,t)\) is continuous from [0, 1] to \(C^0(S^2,M)\cap W^{1,2}(S^2,M)\) in a strong sense. Given a map \(\beta \in \Omega \), the homotopy class \(\Omega _\beta \) is defined to be the set of maps \(\sigma \in \Omega \) that is homotopic to \(\beta \) through maps in \(\Omega \). We’ll call any such \(\sigma \) a sweepout. The width \(W=W_E(\beta ,M)\) associated to the homotopy class \(\Omega _\beta \) is defined as follows:

$$\begin{aligned} W:=\inf _{\sigma \in \Omega _\beta }\max _{t\in [0,1]}E(\sigma (\cdot ,t)). \end{aligned}$$
(23)

We could alternatively define the width using area rather than energy by setting

$$\begin{aligned} W_A:=\inf _{\sigma \in \Omega _\beta }\max _{t\in [0,1]}\text {Area}(\sigma (\cdot ,t)). \end{aligned}$$

Remark 2.16

We’re interested in the case where \(\beta \) induces a map in a non-trivial class in \(\pi _3(M)\), in which case the width is positive.

Definition 2.17

(Minimizing sequence) Given a sweepout \(\gamma _{j}(\cdot ,t):S^2\times [0,1]\rightarrow M,\) we call \(\{\gamma _{j}(\cdot ,\cdot )\}_{j\in {\mathbb {N}}}\) a minimizing sequence if

$$\begin{aligned} \lim \limits _{j\rightarrow \infty }\max _{t\in [0,1]}E(\gamma _j(\cdot ,t))=W. \end{aligned}$$

We call \(\{\gamma _j(\cdot ,t_j)\}_{j\in {\mathbb {N}}}\) a min–max sequence if

$$\begin{aligned} \lim _{j\rightarrow \infty }E(\gamma _j(\cdot ,t_j))=W. \end{aligned}$$

Definition 2.18

We define the equivalent class of \(u:S^2\rightarrow M\) as follows:

$$\begin{aligned}{}[u]:=\Big \{g:S^2\rightarrow M\Big |\text { if } u=g\circ \phi \text { for some }\phi \in PSL(2,{\mathbb {C}})\Big \}, \end{aligned}$$

Remark 2.19

Given maps \(\gamma ,\{u_i\}_{i=0}^n\in W^{1,2}(S^2,M)\) with \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \), we have

$$\begin{aligned} d_B(\gamma ,\{g_i\}_{i=0}^n)<\epsilon ,\,\text {if }[g_i]=[u_i],\,i=0,\ldots ,n. \end{aligned}$$

Definition 2.20

(Image set) The image set \(\Lambda (\{\gamma _j(\cdot ,t)\})\) of \(\{\gamma _j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is defined as follows:

$$\begin{aligned} \Lambda (\{\gamma _j(\cdot ,t)\}_{j\in {\mathbb {N}}}):=\Big \{\{[u_i]\}_{i=0}^n:&\text {there exists a sequence }\{i_j\}\rightarrow \infty , t_{i_j}\in [0,1],\\&\text {such that }\gamma _{i_j}(\cdot ,t_{i_j})\text { bubble converges to }\{u_i\}_{i=0}^n\Big \}, \end{aligned}$$

Now we state the min–max theorem for harmonic sphere. Theorem 2.21 isn’t exactly what’s stated in [1], it uses \(d_B(\cdot ,\cdot )\) instead of varifold norm and applies to any minimizing sequence. We prove in appendix A that Colding–Minicozzi’s result [1] does imply Theorem 2.21.

Theorem 2.21

(Min–Max for harmonic sphere) Given a closed manifold M with dimension at least three, and a map \(\beta \in \Omega \) representing a non-trivial class in \(\pi _3(M)\), then for any sequence of sweepouts \(\gamma _j\in \Omega _\beta \) with

$$\begin{aligned} \lim _{j\rightarrow \infty }\max _{s\in [0,1]}E(\gamma _j(\cdot ,s))= W, \end{aligned}$$

there exists a subsequence \(\{i_j\}\rightarrow \infty \), \(t_{i_j}\in [0,1]\), and a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\) such that

$$\begin{aligned} d_B(\gamma _{i_j}(\cdot ,t_{i_j}),\{u_i\}_{i=0}^n)<1/j. \end{aligned}$$

3 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 \(u\in W^{1,2}(S^2,M)\), and \(X\in C^\infty (S^2,{\mathbb {R}}^N)\) with

$$\begin{aligned} \delta ^2E(u)(X,X)<0. \end{aligned}$$

By the form of second variation of energy (see (10)), clearly if \(\epsilon >0\) is sufficiently small, then for any \(\gamma \in W^{1,2}(S^2,M)\) with \(\Vert \gamma -u\Vert _{W^{1,2}}<\epsilon \), we have

$$\begin{aligned} \delta ^2E(\gamma )(X,X)<0. \end{aligned}$$

Now we consider the general case, given a finite collection of harmonic spheres \(\{u_i\}_{i=0}^n\) with \(\sum _{i=0}^n \text {Index}(u_i)=k>0\). Index assumption implies there are k corresponding vector fields \(\{X_l\}_{l=1}^k\), \(X_l\in C^{\infty }(S^2,{\mathbb {R}}^N)\), positive constant \(c_l>0\) for each l, and the corresponding harmonic spheres \(v_l\in \{u_i\}_{i=0}^n\), such that

$$\begin{aligned} \delta ^2 E(v_l)(X_l,X_l)=-c_l<0. \end{aligned}$$

Now the goal is to choose \(\epsilon >0\) so that for any \(\gamma \in W^{1,2}(S^2,M)\) with \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \), we can construct \(\{{\tilde{X}}_l\}_{l=1}^k\), \({\tilde{X}}_l\in C^\infty (S^2,{\mathbb {R}}^N)\), then the second variation of \(\gamma _s=\Pi \circ \big (\gamma +s{\tilde{X}}_l)\) with respect to \(s\in {\bar{B}}^k\) is negative.

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 \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \) for some \(\gamma \in W^{1,2}(S^2,M)\), since \(v_l\in \{u_i\}_{i=0}^n\), there exist \(\Omega _l\subset S^2\) and \(D_l\in PSL(2,{\mathbb {C}})\), so that

$$\begin{aligned} \int _{\Omega _l}|\nabla \gamma -\nabla (v_l\circ D_l)|^2<\epsilon ^2, \end{aligned}$$

and

$$\begin{aligned} \int _{S^2\setminus \Omega _l}|\nabla (v_l\circ D_l)|^2<\epsilon ^2, \end{aligned}$$
(24)

see Definition (2.8). If \(\epsilon \) is sufficiently small the following term is small

$$\begin{aligned}\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{\Omega _l}|\nabla \Pi \circ (v_l\circ D_l+sX_l\circ D_l)|^2-\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{\Omega _l}|\nabla \Pi \circ (\gamma +sX_l\circ D_l)|^2.\end{aligned}$$

By choosing a suitable cutoff function \(\eta _l\), we can make the following term small

$$\begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2}|\nabla \Pi \circ (\gamma +s\eta _lX_l\circ D_l)|^2-\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{\Omega _l}|\nabla \Pi \circ (\gamma +sX_l\circ D_l)|^2. \end{aligned}$$

Let \({\tilde{X}}_l=\eta _lX_l\circ D_l\) and \(\gamma _s=\Pi \circ \big (\gamma +s{\tilde{X}}_l)\), we observe that

$$\begin{aligned} \begin{aligned} \delta ^2E(v_l)(X,X)&=\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{\Omega _l}|\nabla \Pi \circ (v_l\circ D_l+sX\circ D_l)|^2\\&\quad +\,\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus \Omega _l}|\nabla \Pi \circ (v_l\circ D_l+sX\circ D_l)|^2. \end{aligned} \end{aligned}$$
(25)

The first term of the right hand side of (25) is close to \(\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2}|\nabla \gamma _s|^2\), and the second term is small because of (24). We have the desired inequality

$$\begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2}|\nabla \gamma _s|^2<\frac{1}{2}\delta ^2E(v_l)(X_l,X_l). \end{aligned}$$

We now state and prove the unstable lemma and specify how to choose \(\epsilon >0\) and \(\eta _l.\)

Lemma 3.1

(Unstable lemma) Let M be a closed manifold of dimension at least three, isometrically embedded in \({\mathbb {R}}^N\). Given a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\) with \(\sum _{i=0}^n Index(u_i)=k\), there exist \(1>c_0>0\) and \(\epsilon >0\), so that if \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \) for \(\gamma \in W^{1,2}(S^2,M)\), then we can construct vector fields \(\{{\tilde{X}}_l\}_{l=1}^k\), \({\tilde{X}}_l\in C^{\infty }(S^2,{\mathbb {R}}^N)\) for each l, define the variation \(\gamma _s\) of \(\gamma \) as

$$\begin{aligned} \gamma _s=\Pi \circ \big (\gamma +\sum _{l=1}^k s_l{\tilde{X}}_l\big ),\quad s=(s_1,\ldots ,s_k)\in {\bar{B}}^k, \end{aligned}$$

and let \(E_{\gamma }(s):=E(\gamma _s),\) so that the following hold:

  1. (1)

    \(E_\gamma (s)\) has a unique maximum at \(m_\gamma \in B^k_\frac{c_0}{\sqrt{10}}(0).\)

  2. (2)

    \(\forall s\in {\bar{B}}^k\) we have

    $$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\gamma (s)\le -c_0Id,\ \end{aligned}$$
    (26)

    and

    $$\begin{aligned} E_\gamma (m_\gamma )-\frac{1}{2c_0}|m_\gamma -s|^2\le E_\gamma (s)\le E(m_\gamma )-\frac{c_0}{2}|m_\gamma -s|^2. \end{aligned}$$
    (27)

Proof

First we focus on choosing \(\epsilon >0\) and constructing \(\tilde{X_l}\) so that

$$\begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}E(\Pi \circ (\gamma +s{\tilde{X}}_l))<0, \end{aligned}$$

for all \(0\le s\le 1\).

Index assumption implies there are k corresponding vector fields \(\{X_l\}_{l=1}^k\), \(X_l\in C^{\infty }(S^2,{\mathbb {R}}^N)\), positive constants \(c_l>0\) for each \(l=1,\ldots ,k\), and the corresponding harmonic spheres \(v_l\in \{u_i\}_{i=0}^n\), such that

$$\begin{aligned} \delta ^2 E(v_l)(X_l,X_l)=-c_l<0. \end{aligned}$$
(28)

Let \(\xi :=c_l/C>0\), C is a constant which will be chosen later. By (10), there exists \(\delta (\xi )>0\) depending on \(\{v_l,X_l\}\) such that

$$\begin{aligned} |\delta ^2 E(v_l)(X_l,X_l)-\delta ^2 E(\gamma )(X_l,X_l)|<\xi , \end{aligned}$$
(29)

for all \(\gamma \) with \(\int _{S^2}|\nabla v_l-\nabla \gamma |^2<\delta (\xi )\). We define \(v_{l,s}:=\Pi \circ (v_l+sX_l)\). There exists \(\rho >0\) such that for all \(p\in S^2\) and \(\varrho <\rho \) we have the following:

$$\begin{aligned} -\xi<\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{B_\varrho (p)}|\nabla v_{l,s}|^2<\xi ,\quad \text {and } \int _{B_\varrho (p)}|\nabla X_l|^2<\xi . \end{aligned}$$
(30)

We choose \(J\in {\mathbb {N}}\) so that

$$\begin{aligned} -\frac{1}{\log 1/J}<\xi , \end{aligned}$$
(31)

and define \(\varepsilon _J\) to be \(\min \Big \{\int _{B_{1/J}(p)}|\nabla v_l|^2\Big |\,p\in S^2\Big \},\,i\in {\mathbb {N}},\) note that \(\varepsilon _J\) is strictly positive. We now choose \(\epsilon >0\) to be the constant satisfying the following inequality

$$\begin{aligned} \epsilon ^2<\min \{\varepsilon _J/2,\xi ,\delta (\xi )\}, \end{aligned}$$
(32)

and consider \(\gamma \in W^{1,2}(S^2,M)\) with \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \). Since \(v_l\in \{u_i\}_{i=0}^n\), there is \(\Omega _l\in S^2\) and a conformal dilation \(D_l:S^2\rightarrow S^2\) such that

$$\begin{aligned} \int _{\Omega _{l}}|\nabla \gamma -\nabla (v_l\circ D_{l})|^2<\epsilon ^2, \end{aligned}$$
(33)

and

$$\begin{aligned} \int _{S^2\setminus \Omega _{l}}|\nabla (v_l\circ D_{l})|^2<\epsilon ^2. \end{aligned}$$
(34)

Moreover, we can choose \({\tilde{\Omega }}_l\) with \(\Omega _l\subset {\tilde{\Omega }}_l\) so that

$$\begin{aligned} \int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla \gamma |^2<\epsilon . \end{aligned}$$

Assume that \(S^2\setminus (D_l\circ {\tilde{\Omega }}_l)\) and \(S^2\setminus (D_l\circ \Omega _l)\) are geodesic balls which center at some point \(p\in S^2\), namely, \(S^2\setminus (D_l\circ \Omega _l)=B_r(p)\) and \(S^2\setminus (D_l\circ {\tilde{\Omega }}_l)=B_{r^2}(p)\) for some r. By Eq. (34) we know that

$$\begin{aligned} \int _{B_r(p)}|\nabla v_l|^2<\varepsilon _J/2, \end{aligned}$$

and it implies that r must be smaller than 1/J.

Now we define the following piecewise smooth cutoff function, which was introduced by Choi and Schoen [3], \(\eta :[0,\infty )\rightarrow [0,1]\):

$$\begin{aligned} \eta (x) =\left\{ \begin{array}{lll} &{}0, \text{ for } x<r^2, \\ &{}-(\log x)/(\log r), \text{ for } r^2\le x \le r,\\ &{}1, \text{ for } x>r, \end{array}\right. \end{aligned}$$

so that

$$\begin{aligned} \frac{d\eta }{dx}(x) =\left\{ \begin{array}{lll} &{}0, \text{ for } x<r^2, \\ &{}-1/x(\log r), \text{ for } r^2\le x \le r,\\ &{}0, \text{ for } x>r, \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{2\pi }\int _{r^2}^{r}\Big (\frac{d\eta }{dx}(x)\Big )^2xdxd\theta =-\frac{2\pi }{\log r}. \end{aligned}$$

Since we have \(r<1/J\), (31) implies that

$$\begin{aligned} -\frac{2\pi }{\log r}<2\pi \xi . \end{aligned}$$

Now we define \(\eta _l=\eta \circ y_l:S^2\rightarrow [0,1]\), where \(y_l:S^2\rightarrow [0,\infty )\) and \(y_l(q)=x\) for \(q\in \partial B_x(p)\). \(\eta _l\) is compactly supported in \(S^2\setminus B_{r^2}(p)\) and has value 1 in \(S^2\setminus B_{r}(p)\), then

$$\begin{aligned} \int _{S^2}|\nabla \eta _l|^2<2\pi \xi . \end{aligned}$$
(35)

Let \({\tilde{X}}'_l:=(\eta _l\circ D_l)X_l\circ D_l,\) and define \(v_{l,s}=\Pi \circ (v_l+sX_l)\), \(\gamma _{s}:=\Pi \circ (\gamma +s{\tilde{X}}_l')\). Then we have the following:

$$\begin{aligned}&\Big |\delta ^2E(\gamma )({\tilde{X}}'_l,{\tilde{X}}'_l)-\delta ^2E(v_l)(X_l,X_l)\Big |\nonumber \\&\quad =\Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2}|\nabla \gamma _s|^2-\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2}|\nabla v_{l,s}|^2\Big |\nonumber \\&\quad <\Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{\Omega _l}|\nabla \gamma _{s}|^2-\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{D_l\circ \Omega _l}|\nabla v_{l,s}|^2\Big |\nonumber \\&\quad \quad +\,\Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus (D_l\circ \Omega _l)}|\nabla v_{l,s}|^2\Big |\nonumber \\&\quad \quad +\,\Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus \Omega _l}|\nabla \gamma _{s}|^2\Big |. \end{aligned}$$
(36)

Since \(\int _{\Omega _l}|\nabla v_l-\nabla \gamma |^2<\epsilon ^2<\delta (\xi )\), (29) implies that

$$\begin{aligned} \Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\Big (\int _{\Omega _l}|\nabla \gamma _{s}|^2-|\nabla (v_{l,s}\circ D_l)|^2\Big )\Big | <\xi , \end{aligned}$$
(37)

Since \(S^2\setminus (D_l\circ \Omega _l)=B_r(p)\) by the assumption, the choice of \(\epsilon \) implies that \(r<\rho \) (see (32)) and Eq. (30) implies that

$$\begin{aligned} \Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus (D_l\circ \Omega _l)}|\nabla v_{l,s}|^2\Big |<\xi . \end{aligned}$$
(38)

Now we consider the last term of Eq. (36), namely:

$$\begin{aligned} \Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus \Omega _l}|\nabla \gamma _{s}|^2\Big |, \end{aligned}$$

by Eq. (10) we have:

$$\begin{aligned} \begin{aligned}&\Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus \Omega _l}|\nabla \gamma _{s}|^2\Big |\\&\quad \le \int _{S^2\setminus \Omega _l}|\mathrm{{d}}\Pi _\gamma (\nabla {\tilde{X}}'_l)|^2+C_1\int _{S^2\setminus \Omega _l}\big (|\nabla {\tilde{X}}_l'||\nabla \gamma |+|{\tilde{X}}_l'|^2|\nabla \gamma |^2\big )\\&\quad \le \int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla {\tilde{X}}_l'|^2\\&\quad \quad +\,C_1\Big (\int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla {\tilde{X}}_l'|^2\Big )^{1/2}\Big (\int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla \gamma |^2\Big )^{1/2}\\&\quad \quad +\,C_1\sup _{p\in S^2}|X_l(p)|^2\int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla \gamma |^2, \end{aligned} \end{aligned}$$
(39)

here \(C_1\) is a constant which depends on M (since \(\text {Hess}\Pi _\gamma (\cdot ,\cdot )\) is bounded by the second fundamental form of M [17, Appendix 2.12] and \(\nabla \text {Hess}\Pi \) is bounded by curvature of M). Then

$$\begin{aligned} \int _{{\tilde{\Omega }}_l\setminus \Omega _l}|\nabla {\tilde{X}}_l'|^2&=\int _{B_{r}(p)\setminus B_{r^k}(p)}|\nabla (\eta _l X_l)|^2\nonumber \\&=\int _{B_{r}(p)\setminus B_{r^k}(p)}|(\nabla \eta _l)X_l+\eta _l\nabla X_l|^2\nonumber \\&\le \int _{B_{r}(p)\setminus B_{r^k}(p)}|\nabla X_l|^2 +\,2\sup _{p\in S^2}|X_l(p)|^2\nonumber \\&\quad \times \Big (\int _{B_{r}(p)\setminus B_{r^k}(p)}|\nabla X_l|^2\Big )^{1/2}\Big (\int _{B_{r}(p)\setminus B_{r^k}(p)}|\nabla \eta _l|^2\Big )^{1/2}\nonumber \\&\quad +\,\sup _{p\in S^2}|X_l(p)|^2\int _{B_{r}(p)\setminus B_{r^k}(p)}|\nabla \eta _l|^2\nonumber \\&<C_2\xi , \end{aligned}$$
(40)

for some constant \(C_2(M,X_l)\), the last inequality follows from Eq. (35) and (30). By (40) we can bound (39) by

$$\begin{aligned} \Big |\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\int _{S^2\setminus \Omega _l}|\nabla \gamma _{s}|^2\Big |<C_2\xi +C_1\sqrt{C_2}\xi +C_1\sup _{p\in S^2}|X_l(p)|^2\xi <C_3\xi . \end{aligned}$$
(41)

Finally, combining the inequality (37), (38), and (41), we have that (36) is bounded by

$$\begin{aligned} \Big |\delta ^2E(\gamma )({\tilde{X}}'_l,{\tilde{X}}'_l)-\delta ^2E(v_l)(X_l,X_l)\Big |<\xi +\xi +C_3\xi . \end{aligned}$$
(42)

Since \(\xi =c_l/C\), we now pick C to be a constant strictly larger than \(5(C_3+2)\), then we have

$$\begin{aligned} -\frac{6}{5}c_l<\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}E_{\gamma }(s)<-\frac{4}{5}c_l, \end{aligned}$$

since \(\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=t}E_{\gamma }(s)\) is continuous with respect to t, there exists \(\kappa _l(M,X_l,u_l,\epsilon )>0\) such that

$$\begin{aligned} -\frac{6}{5}c_l<\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=t}E_{\gamma }(s)<-\frac{4}{5}c_l,\quad \text {for all }t\in [-\kappa _l,\kappa _l]. \end{aligned}$$

We can choose a constant \(a(\kappa _l)>0\), let \({\tilde{X}}_l:=a(\kappa _l)(\eta _l\circ D_l)X_l\circ D_l,\) and redefine the variation of \(\gamma \) to be \(\gamma _{s}:=\Pi \circ (\gamma +s{\tilde{X}}_l)\) so that

$$\begin{aligned} -a^2(\kappa _l)\frac{3}{2}c_l<\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=t}E_{\gamma }(s)<-a^2(\kappa _l)\frac{1}{2}c_l,\quad \text {for all }t\in [-1,1]. \end{aligned}$$

By choosing a constant \(0<c_0<1\) such that \(c_0<\min _{l=1,..,k}a^2(\kappa _l)\frac{1}{2}c_l\) and \(1/c_0>\min _{l=1,\ldots ,k}a^2(\kappa _l)\frac{3}{2}c_l\), we get

$$\begin{aligned} -\frac{1}{c_0}<\frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=t}E_{\gamma }(s)<-c_0\quad \text {for all }t\in [-1,1]. \end{aligned}$$

In order to show the following

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\gamma (s)\le -c_0Id, \end{aligned}$$

we need to check that

$$\begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=t}E(\Pi \circ (\gamma +s({\tilde{X}}_i+{\tilde{X}}_j))). \end{aligned}$$
(43)

is negative for all \(0\le t\le 1\) and i, j ranges from 1 to k. There are two possible cases, one is vector fields \(X_i\) and \(X_j\) both contribute index to the same harmonic sphere \(v\in \{u_i\}_{i=0}^n\). That is,

$$\begin{aligned} \delta ^2E(v)(Y,Y)<0,\quad \forall Y=aX_i+bX_j,\,a,b\in {\mathbb {R}}. \end{aligned}$$

In this case, it’s obvious that (43) is negative. In the other case, if \(X_i\) \(X_j\) contribute index to different harmonic sphere \(v_i,v_j\in \{u_i\}_{i=0}^n\), then we obviously have

$$\begin{aligned} \frac{\mathrm{{d}}^2}{\mathrm{{d}}s^2}\Big |_{s=0}\Big (E(\Pi \circ (v_i+sX_i)+E(\Pi \circ (v_j+sX_j))\Big )<0. \end{aligned}$$
(44)

Since

$$\begin{aligned} \begin{aligned} \delta ^2E(\gamma )({\tilde{X}}_i+{\tilde{X}}_j,{\tilde{X}}_i+{\tilde{X}}_j)&=\delta ^2E(\gamma )({\tilde{X}}_i,{\tilde{X}}_i)+\delta ^2E(\gamma )({\tilde{X}}_j,{\tilde{X}}_j)\\&\quad +\,2\delta ^2E(\gamma )({\tilde{X}}_i,{\tilde{X}}_j), \end{aligned} \end{aligned}$$
(45)

thus if \(\delta ^2E(\gamma )({\tilde{X}}_i,{\tilde{X}}_j)\) is sufficiently small then the left hand side of Eq. (45) is negative. From (10) we have

$$\begin{aligned} \begin{aligned} \delta ^2 E(\gamma )({\tilde{X}}_i,{\tilde{X}}_j)&=2\int _{S^2}\langle \mathrm{{d}}\Pi _\gamma (\nabla {\tilde{X}}_i),\mathrm{{d}}\Pi _\gamma (\nabla {\tilde{X}}_j)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _\gamma ({\tilde{X}}_i,\nabla \gamma ),\mathrm{{d}}\Pi _\gamma (\nabla {\tilde{X}}_j)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _\gamma ({\tilde{X}}_j,\nabla \gamma ),\mathrm{{d}}\Pi _\gamma (\nabla {\tilde{X}}_i)\rangle \\&\quad +\,2\int _{S^2}\langle \text {Hess}\Pi _\gamma ({\tilde{X}}_i,\nabla \gamma ),\text {Hess}\Pi _\gamma ({\tilde{X}}_j,\nabla \gamma )\rangle \\&\quad +\,2\int _{S^2}\langle \nabla u,\big ( 2\text {Hess}\Pi _\gamma ({\tilde{X}}_i,\nabla {\tilde{X}}_j)+\nabla \text {Hess}\Pi _\gamma ({\tilde{X}}_i,{\tilde{X}}_j,\nabla \gamma )\big )\rangle . \end{aligned} \end{aligned}$$
(46)

We recall from the construction of \({\tilde{X}}_i\) that the support of \({\tilde{X}}_i\) and \({\tilde{X}}_j\) intersect at \(S^2\setminus (\Omega _i\bigcup \Omega _j)\), so we only need to consider the right hand side of (46) integrating on \(S^2\setminus (\Omega _i\bigcup \Omega _j)\). We argue similarly as before using the choice of the cutoff functions \(\eta _i,\eta _j\) and \(\int _{S^2\setminus (\Omega _i\bigcup \Omega _j)}|\nabla \gamma |^2<\epsilon \) to show that we can make \(\delta ^2 E(\gamma )({\tilde{X}}_i,{\tilde{X}}_j)\) sufficiently small. We omit the details of estimates here since it’s similar as what we did before. Thus we complete the proof of

$$\begin{aligned}-\frac{1}{c_0}Id\le D^2E_\gamma (s)\le -c_0Id.\end{aligned}$$

We’re left to show that \(E_\gamma (s)\) has a unique maximum at \(m_\gamma \in B^k_\frac{c_0}{\sqrt{10}}(0).\) With fixed \(\gamma \) and \(\{{\tilde{X}}_l\}_{l=1}^k\), \(E_\gamma (s)\) is a smooth function with respect to \(s\in {\bar{B}}^k\). Again, we can argue similarly to show that \(|DE_\gamma (s)|\) is sufficiently small. Thus combining with that \(E_\gamma (s)\) is concave we get \(E_\gamma (s)\) has a unique ( for the fixed \(\{{\tilde{X}}_l\}_{l=1}^k\) we already chose) maximum at \(m_\gamma \in B^k_\frac{c_0}{\sqrt{10}}(0)\). We omit the details of estimates here since it’s similar as the method used before. \(\square \)

The vector fields \(\{{\tilde{X}}_l\}_{l=1}^k\) constructed in Lemma 3.1 depend on \(\gamma \). We are about to show that in the following corollary there exists \(\delta _\gamma >0\) such that for all maps \(\sigma \) with \(\Vert \sigma -\gamma \Vert _{W^{1,2}}<\delta _\gamma \). The variation of \(\sigma \) with respect to vector fields \(\{{\tilde{X}}_l\}_{l=1}^k\) constructed with respect to \(\gamma \) still satisfies (26) and (27) in Lemma 3.1.

Corollary 3.2

Let M be a closed manifold of dimension at least three, isometrically embedded in \({\mathbb {R}}^N\). Given a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\) with \(\sum _{i=0}^n Index(u_i)=k\). Let \(1>c_0>0\) and \(\epsilon >0\) be given by Lemma 3.1. For a map \(\gamma \in W^{1,2}(S^2,M)\) with \(d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \), let \(\{{\tilde{X}}_l\}_{l=1}^k\) be vector fields given as Lemma 3.1. For \(\sigma \in W^{1,2}(S^2,M)\), we define

$$\begin{aligned} \sigma _s:=\Pi \circ (\sigma +\sum _{l=1}^ks_l{\tilde{X}}_l)\quad \text {for }s=(s_1,\ldots ,s_k)\in {\bar{B}}^k, \end{aligned}$$

and let \(E_\sigma (s):=E(\sigma _s)\), \(m_\sigma \) be the maximum of \(E_\sigma (s)\). There exists \(\delta _\gamma >0\), such that for \(\sigma \in W^{1,2}(S^2,M)\) satisfying

$$\begin{aligned} \int _{S^2}|\nabla \gamma -\nabla \sigma |^2<\delta _\gamma , \end{aligned}$$

the following properties hold

  1. (1)

    \(E_\sigma (s)\) has a unique maximum at \(m_\sigma \in B^k_\frac{c_0}{\sqrt{10}}(0).\)

  2. (2)

    \(\forall s\in {\bar{B}}^k\) we have

    $$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\sigma (s)\le -c_0Id,\ \end{aligned}$$
    (47)

    and

    $$\begin{aligned} E_\sigma (m_\sigma )-\frac{1}{2c_0}|m_\sigma -s|^2\le E_\sigma (s)\le E_\sigma (m_\sigma )-\frac{c_0}{2}|m_\sigma -s|^2. \end{aligned}$$
    (48)

Proof

By Lemma 3.1, \(\forall s\in {\bar{B}}^k\), we have

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\gamma (s)\le -c_0Id. \end{aligned}$$
(49)

For each \(s\in {\bar{B}}^k\), there exists \(\delta '(s)>0\) so that for all \(\sigma \in W^{1,2}(S^2,M)\) with

$$\begin{aligned} \int _{S^2}|\nabla \sigma -\nabla \gamma |^2<\delta '(s), \end{aligned}$$

we have

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\sigma (s)\le -c_0Id. \end{aligned}$$
(50)

We define

$$\begin{aligned} \delta (s):=\frac{1}{2}\max \Big \{\delta '(s)>0\Big |-\frac{1}{c_0}Id&\le D^2E_\sigma (s)\le -c_0Id,\\&\text { if }\int _{S^2}|\nabla \sigma -\nabla \gamma |^2<\delta '(s)\Big \}. \end{aligned}$$

From Eqs. (6) and (7) we can see that \(\delta (s)\) is continuous with respect to s. Let \(\delta _\gamma :=\inf _{s\in \bar{B}^k}\delta (s)\).

Claim 3.3

\(\delta _\gamma >0\).

Proof of the claim

If not, there exists a sequence \(\{s_i\}_{i\in {\mathbb {N}}}\) such that \(\lim _{i\rightarrow \infty }\delta (s_i)=0\). Since \({\bar{B}}^k\) is compact, we have that \(\lim _{i\rightarrow \infty }s_i=s'\in {\bar{B}}^k\), and \(\delta (s')>0\) implies the desired contradiction. \(\square \)

Thus, \(\forall \sigma \in W^{1,2}(S^2,M)\) with

$$\begin{aligned} \int _{S^2}|\nabla \sigma -\nabla \gamma |^2<\delta _\gamma , \end{aligned}$$

we have

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2E_\sigma (s)\le -c_0Id,\quad \forall s\in {\bar{B}}^k. \end{aligned}$$
(51)

Again, we can argue similarly to show that we can choose \(\delta _\gamma >0\) to be sufficiently small so that for all maps \(\sigma \in W^{1,2}(S^2,M)\) satisfying

$$\begin{aligned} \int _{S^2}|\nabla \sigma -\nabla \gamma |^2<\delta _\gamma , \end{aligned}$$

we have that \(|DE_\sigma (s)|\) is sufficiently small. Thus, combining with that \(E_\sigma (s)\) is concave we get that \(E_\sigma (s)\) has an unique maximum at \(m_\sigma \in B^k_\frac{c_0}{\sqrt{10}}(0).\) \(\square \)

Let M be a closed manifold of dimension at least three, isometrically embedded in \({\mathbb {R}}^N\). Given a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\), \(u_i:S^2\rightarrow M\), with \(\sum _{i=0}^n Index(u_i)=k\). Index assumption implies that there are k correspongding vector fields \(\{X_l\}_{l=1}^k\), \(X_l\in C^{\infty }(S^2,{\mathbb {R}}^N)\), positive constants \(c_l>0\) for each \(l=1,\ldots ,k\), and the corresponding harmonic spheres \(v_l\in \{u_i\}_{i=0}^n\), such that

$$\begin{aligned} \delta ^2 E(v_l)(X_l,X_l)=-c_l<0.\quad l=1,\ldots ,k. \end{aligned}$$
(52)

Lemma 3.4

As assumed above, there exists a constant \(\varsigma >0\), which depends on \(\big \{\{u_i\}_{i=0}^n,\{X_l\}_{l=1}^k\big \}\), so that for \(\gamma \in W^{1,2}(S^2,M)\), if \(\epsilon /2<d_B(\gamma ,\{u_i\}_{i=0}^n)<\epsilon \) (the constant \(\epsilon >0\) is given by Lemma 3.1), and

$$\begin{aligned} E\left( \Pi \circ (\gamma +\sum _{l=1}^ks_l{\tilde{X}}_l)\right) \le E(\gamma )+\varsigma ,\quad (s_1,\ldots ,s_k)\in {\bar{B}}^k, \end{aligned}$$

where the vector fields \(\{{\tilde{X}}_l\}_{l=1}^k\) are given by Lemma 3.1), depending on \(\big \{\gamma ,\{u_i\}_{i=0}^n,\{X_l\}_{l=1}^k\big \}\), then we have

$$\begin{aligned} d_B\left( \Pi \circ (\gamma +\sum _{l=1}^ks_l{\tilde{X}}_l),\{u_i\}_{i=0}^n\right) >2\varsigma . \end{aligned}$$

Proof

We argue by contradiction. Assuming there is a sequence of maps \(\gamma _j\in W^{1,2}(S^2,M)\), \(\epsilon /2<d_B(\gamma _j,\{u_i\}_{i=0}^n)<\epsilon \) for all \(j\in {\mathbb {N}}\),

$$\begin{aligned} E\left( \Pi \circ \left( \gamma _j+\sum _{l=1}^ks^j_l{\tilde{X}}^j_l\right) \right) \le E(\gamma _j)+1/j, \end{aligned}$$
(53)

here \(\{{\tilde{X}}^j_l\}_{l=1}^k\) are the vector fields given by Lemma 3.1 for each \(\gamma _j\)

$$\begin{aligned} d_B\left( \Pi \circ (\gamma _j+\sum _{l=1}^ks^j_l{\tilde{X}}^j_l),\{u_i\}_{i=0}^n\right) \le 2/j. \end{aligned}$$
(54)

Since we have \(d_B(\gamma _j,\{u_i\}_{i=0}^n)<\epsilon \), there exist conformal dilations \(\{D^j_{i}\}_{i=0}^n,\) and pairwise disjoint domains \(\Omega ^j_{0},\ldots ,\Omega ^j_{n}\), \(\bigcup _{i=0}^n\Omega ^j_{i}\subset S^2\) so that the following holds:

$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{\Omega ^j_{i}}|\nabla \gamma _j-\nabla (u_i\circ D^j_{i})|^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(55)
$$\begin{aligned}&\Big (\int _{S^2\setminus \bigcup _{i=0}^n\Omega _{i}^j}|\nabla \gamma _j|^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(56)
$$\begin{aligned}&\sum _{i=0}^n\Big (\int _{S^2\setminus \Omega _{i}^j}|\nabla (u_i\circ D_{i}^j)|^2\Big )^{1/2}<\epsilon , \end{aligned}$$
(57)

for all \(j\in {\mathbb {N}}.\) Since \(v_l\in \{u_i\}_{i=0}^n\), the Assumption 54 implies that

$$\begin{aligned} \int _{\Omega ^j_{l}}\left| \nabla \big (\Pi \circ (\gamma _j+\sum _{l=1}^ks^j_l{\tilde{X}}^j_l)\big )-\nabla (v_l\circ D^j_{l})\right| ^2<1/j. \end{aligned}$$
(58)

We abuse the notation here by assuming \(\Omega _l^j\) and \(D^j_l\) are the corresponding domain and conformal dilation for \(v_l\in \{u_i\}_{i=0}^n\) such that

$$\begin{aligned} \Big (\int _{\Omega ^j_l }|\nabla \gamma _j-\nabla (v_l\circ D^j_{l})|^2\Big )^{1/2}<\epsilon ,\text { and } \Big (\int _{S^2\setminus \Omega _{l}^j}|\nabla (v_l\circ D_l^j)|^2\Big )^{1/2}<\epsilon . \end{aligned}$$

Since by the construction of \(\{{\tilde{X}}_l^j\}_{l=1}^k\) in Lemma 3.1 we know that each \({\tilde{X}}_l^j\) is supported on \(\Omega ^l_i\), thus (58) becomes \(\int _{\Omega ^j_{l}}|\nabla \big (\Pi \circ (\gamma _j+s^j_l{\tilde{X}}^j_l)\big )-\nabla (v_l\circ D^j_{l})|^2<1/j,\) and we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\Pi \circ (\gamma _j+s^j_l{\tilde{X}}^j_l)\circ (D^j_l)^{-1}(x)=v_l(x),\quad \text {for almost every } x\in S^2. \end{aligned}$$

This implies that for almost every \(x\in S^2\) we have

$$\begin{aligned} \lim _{j\rightarrow \infty }(\gamma _j+s^j_l{\tilde{X}}^j_l)\circ (D^j_l)^{-1}(x)=v_l(x)+{\tilde{\nu }}_l(x), \end{aligned}$$

for some \({\tilde{\nu }}_l(x)\in T_{v_l(x)}^{\bot }M\). Let \(Y^j_l(x):=\gamma _j\circ (D^j_l)^{-1}(x)-v_l(x)\), then

$$\begin{aligned} \lim _{j\rightarrow \infty }\mathrm{{d}}\Pi _{v_l}(-Y^j_l)(x)&=\lim _{j\rightarrow \infty }\mathrm{{d}}\Pi _{v_l}\big (s^j_l{\tilde{X}}^j_l\circ (D^j_l)^{-1}(x)-{\tilde{\nu }}_l(x)\big )\\&=\lim _{j\rightarrow \infty }\mathrm{{d}}\Pi _{v_l}\big (s^j_l{\tilde{X}}^j_l\circ (D^j_l)^{-1}(x))\\&=\mathrm{{d}}\Pi _{v_l}(s_lX_l), \end{aligned}$$

where \(s_l:=\lim _{j\rightarrow \infty }s^j_l\), the second equality follows from [17, Chapter 2.12.3], and the third equality follows from the construction of \({\tilde{X}}_l^j\) in Lemma 3.1. So we have that

$$\begin{aligned} \lim _{j\rightarrow \infty }\delta ^2E(v_l)(-Y^j_l,-Y^j_l)=\delta ^2E(v_l)(s_lX_l,s_lX_l)\le 0. \end{aligned}$$
(59)

If \(s\ne 0\) then the inequality of (59) is strictly negative. (59) implies that

$$\begin{aligned} \lim _{j\rightarrow \infty }E(\gamma _j)&=\lim _{j\rightarrow \infty }\sum _{l=1}^k\int _{\Omega _l^j}|\nabla (\Pi \circ (v_l\circ D^j_l+(\gamma _j-v_l\circ D^j_l))|^2\\&=\lim _{j\rightarrow \infty }\sum _{l=1}^k\int _{\Omega _l^j\circ (D^j_l)^{-1}}|\nabla (\Pi \circ (v_l+Y_l^j))|^2\\&\le \sum _{i=0}^nE(u_i), \end{aligned}$$

where the last inequality follows from (59), and the equality holds if and only if \(s_l=0\) for \(l=1,\ldots ,k\). On the other hand, by assumption (53), we have

$$\begin{aligned} \sum _{i=0}^nE(u_i)=\lim _{j\rightarrow \infty }E\left( \Pi \circ (\gamma _j+\sum _{l=1}^ks^j_l{\tilde{X}}^j_l)\right) \le \lim _{j\rightarrow \infty } E(\gamma _j), \end{aligned}$$

which forces \(s_l\) to be 0 for \(l=1,\ldots ,k\). Thus (54) implies that \(\lim _{j\rightarrow \infty }d_B(\gamma _j,\{u_i\}_{i=0}^n)=0\), which is the desired contradiction. \(\square \)

4 Deformation Theorem

Let M be a closed manifold with dimension at least three, isometrically embedded in \({\mathbb {R}}^N\). Consider a map \(\beta \in \Omega \) representing a non-trivial class in \(\pi _3(M)\), let W be the width associated to the homotopy class \(\Omega _\beta \) (see Definition 2.15, (23)), and given a sequence of sweepouts \(\gamma _{j}(\cdot ,t)\in \Omega _\beta \) which is minimizing, i.e.,

$$\begin{aligned} \lim _{j\rightarrow \infty }\max _{t\in [0,1]}E(\gamma _j(\cdot ,t))=W. \end{aligned}$$

Moreover, let \(K=\Big \{\{[k^1_i]\}_{i=0}^{m_1},\ldots ,\{[k^{N_k}_i]\}_{i=0}^{m_{N_k}}\Big \}\) be a finite set of finite collection of equivalent classes of harmonic spheres, so that there exist a constant \(\epsilon _k>0\) and \(j_k\in {\mathbb {N}}\) such that

$$\begin{aligned} d_B(\gamma _j(\cdot ,t),\{k^l_i\}_{i=0}^{m_l})>\epsilon _k,\quad \forall t\in [0,1], \end{aligned}$$

for all \(j>j_k\), \(l=1,\ldots ,N_k\).

Theorem 4.1

(Deformation Theorem) As assumed above, given a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^n\) with \(\sum _{i=0}^n \text {Index}(u_i)=k>1\) and \(\sum _{i=0}^nE(u_i)=W\), there exists a sequence of sweepouts \(\{\gamma _j'(\cdot ,t)\}_{j\in {\mathbb {N}}}\) such that

  1. (1)

    \(\gamma '_j(\cdot ,t)\) is homotopic to \(\gamma _j(\cdot ,t)\),

  2. (2)

    \(\{\gamma '_j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is a minimizing sequence,

  3. (3)

    there exists \(j_k'\in {\mathbb {N}}\) such that

    $$\begin{aligned} d_B(\gamma '_j(\cdot ,t),\{k^l_i\}_{i=0}^{m_l})>\epsilon _k\quad \text {for }l=1,\ldots ,N_k,\forall t\in [0,1], \end{aligned}$$

    for all \(j>j_k'\).

  4. (4)

    there exists \(\epsilon _J>0\) and \(J\in {\mathbb {N}}\) such that

    $$\begin{aligned} d_B(\gamma _j'(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _J,\quad \forall t\in [0,1], \end{aligned}$$

    for all \(j>J\).

Proof

Let

$$\begin{aligned} I_{j,\epsilon }:=\Big \{t\in [0,1]\big |d_B(\gamma _j(\cdot ,t),\{u_i\}_{i=0}^n)<\epsilon \Big \}, \end{aligned}$$

here the constant \(\epsilon >0\) is given by Lemma 3.1. We define

$$\begin{aligned} E_j^t(s,\{Y_l\}_{l=1}^k):=\int _{S^2}\left| \nabla (\Pi \circ (\gamma _j(\cdot ,t)+\sum _{l=1}^ks_lY_l))\right| ^2,\quad s=(s_1,\ldots ,s_k)\in {\bar{B}}^k. \end{aligned}$$

For \(t_m\in I_{j,\epsilon }\), by Lemma 3.1, we can construct vector fields \(\{{\tilde{X}}_l(t_m)\}_{l=1}^k\), and the hessian of \(E_j^{t_m}(s,\{{\tilde{X}}_l(t_m)\}_{l=1}^k)\) with respect to \(s\in {\bar{B}}^k\), which we denote by \(D^2_sE_j^{t_m}(s,\{{\tilde{X}}_l(t_m)\}\), satisfies

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2_sE_j^{t_m}(s,\{{\tilde{X}}_l(t_m)\}_{l=1}^k)\le -c_0Id,\,\forall s\in {\bar{B}}^k \end{aligned}$$
(60)

here \(c_0\) is a constant given by Lemma 3.1. By Corollary 3.2 and the continuity of \(\gamma _j(\cdot ,t)\) in \(W^{1,2}(S^2,M)\) with respect to t, we know there exists \(\delta (t_m)>0\) so that for all \( t\in (t_m-\delta (t_m),t_m+\delta (t_m))\cap I_{j,\epsilon }\) we have

$$\begin{aligned} -\frac{1}{c_0}Id\le D^2_sE_j^{t}(s,\{{\tilde{X}}_l(t_m)\}_{l=1}^k)\le -c_0Id,\,\forall s\in {\bar{B}}^k. \end{aligned}$$
(61)

Let \(I^{t_m}:=(t_m-\delta (t_m),t_m+\delta (t_m))\cap I_{j,\epsilon }\), since \({\bar{I}}_{j,\epsilon }\) is compact we can cover \({\bar{I}}_{j,\epsilon }\) by finitely many \(I^{t}\), say \(I^{t_1},\ldots ,I^{t_{N_1}}\). Moreover, after discarding some of the intervals, we can arrange that each t is in at least one closed interval \(\bar{I}^{t_m}\), each \(\bar{I}^{t_m}\) intersects at most two other \(\bar{I}^{t_k}\)’s, and the \(\bar{I}^{t_k}\)’s intersecting \(\bar{I}^{t_m}\) do not intersect each other. For each \(m=1,\ldots ,N_1\), choose a smooth function \(\xi _m(t):[0,1]\rightarrow [0,1]\) which is supported in \(\bar{I}^{t_m}\), and

$$\begin{aligned} \sum _{m=1}^{N_1}\xi _m(t)=1,\,\forall t\in [0,1]. \end{aligned}$$

We define \(X_l(t)\) to be

$$\begin{aligned} X_l(t):=\sum _{m=1}^{N_1}\xi _m(t){\tilde{X}}_l(t_m),\quad t\in [0,1],l=1,\ldots ,k. \end{aligned}$$
(62)

Consider \(E_j^{t}(s,\{{X}_l(t)\}_{l=1}^k)\), if \(X_l(t)=\tilde{X}_l(t_m)\) for some \(t\in I_{j,\epsilon }\), then obviously we have \(-\frac{1}{c_0}Id\le D^2_sE_j^{t}(s,\{{X}_l(t)\}_{l=1}^k)\le -c_0Id,\,\forall s\in {\bar{B}}^k.\) If \(X_l(t)=\delta _a(t)\tilde{X}_l(t_a)+\delta _b(t)\tilde{X}_l(t_b)\), since \(\delta _a(t)+\delta _b(t)=1\), we have for all \(s\in \bar{B}^k\)

$$\begin{aligned} \begin{aligned} -\frac{2}{c_0}Id&<D^2_sE_j^{t}(s,\{{X}_l(t)\}_{l=1}^k)\\&=D^2_sE_j^{t}(s,\{\delta _a(t)\tilde{X}_l(t_a)+\delta _b(t)\tilde{X}_l(t_b)\}_{l=1}^k)\\&<\delta ^2_a(t)D^2_sE_j^{t}(s,\{\tilde{X}_l(t_a)\}_{l=1}^k)+\delta ^2_b(t)D^2_sE_j^{t}(s,\{\tilde{X}_l(t_b)\}_{l=1}^k)\\&\le -\frac{c_0}{2}Id. \end{aligned} \end{aligned}$$
(63)

The last inequality follows from \(\delta ^2_a(t)+\delta ^2_b(t)\ge 1/2\). By (63), we can choose \(c=c_0/2\) such that \(-\frac{1}{c}Id<D^2_sE_j^{t}(s,\{{X}_l(t)\}_{l=1}^k)<-cId,\,\forall s\in \bar{B}^k.\) Now we define

$$\begin{aligned} \gamma _{j,s}(\cdot ,t):=\Pi \circ \Big (\gamma _j(\cdot ,t)+\sum _{l=1}^ks_lX_{l}(t)\Big ),\quad s=(s_1,..,.s_k)\in {\bar{B}}^{k}, \end{aligned}$$

and let \(E_j^t(s):=E(\gamma _{j,s}(\cdot ,t)),\) then we have

  1. (1)

    \(E_j^t(s)\) has a unique maximum at \(m_j(t)\in B^k_\frac{c}{\sqrt{10}}(0).\)

  2. (2)

    The map \(\gamma _j(\cdot ,t)\mapsto m_j(t)\) is continuous.

  3. (3)

    \(\forall s\in {\bar{B}}^k\) and \(\forall t\in I_{j,\epsilon }\) we have

    $$\begin{aligned} -\frac{1}{c}Id\le D^2E_j^t(s)\le -c Id, \end{aligned}$$
    (64)

    and

    $$\begin{aligned} E_j^t(m_j(t))-\frac{1}{2c}|m_j(t)-s|^2\le E^t_j(s)\le E_j^t(m_j(t))-\frac{c}{2}|m_j(t)-s|^2. \end{aligned}$$
    (65)

Recall \(\forall \{[k^l_i]\}_{i=0}^{m_l}\in K\) we have: \(d_B(\gamma _j(\cdot ,t),\{k^l_i\}_{i=0}^{m_l})>\epsilon _l\) for \(l=1,\ldots ,N_k,\forall t\in [0,1].\) for all \(j>j_k\). Without loss of generality (by rescaling \(\{X_l(t)\}_{l=1}^k\) and c), we can assume that there exists \(j_k'\in {\mathbb {N}}\) such that

$$\begin{aligned} d_B(\gamma _{j,s}(\cdot ,t),\{k^l_i\}_{i=0}^{m_l})>\epsilon _l\quad \text {for }l=1,\ldots ,N_k,\forall s\in {\bar{B}}^k,\forall t\in [0,1], \end{aligned}$$
(66)

for all \(j>j'_k\).

Now we consider the one-parameter flow

$$\begin{aligned} \{\phi _j^t(\cdot ,x)\}_{x\ge 0}&\in \text {Diff}({\bar{B}}^k)\\ \phi _j^t(\cdot ,\cdot ):{\bar{B}}^k&\times [0,\infty )\rightarrow {\bar{B}}^k, \end{aligned}$$

generated by the vector field:

$$\begin{aligned} s\mapsto -(1-|s|^2)\nabla E_j^t(s),\,s\in {\bar{B}}^k. \end{aligned}$$
(67)

Claim 4.2

For all \(\kappa <\frac{1}{4}\), there is \(T_j\) depending on \(\big \{\{u_i\}_{i=0}^n,\{X_l\}_{l=1}^k,\kappa ,\epsilon \big \}\) so that for any \(t\in I_{j,\epsilon }\), and \(v\in {\bar{B}}^k\) with \(|v-m_j(t)|\ge \kappa \) we have:

$$\begin{aligned} E_j^t(\phi _j^t(v,T_j))<E_j^t(0)-\frac{c}{10}. \end{aligned}$$
(68)

Proof

By \(m_j(t)\in B^k_\frac{c}{\sqrt{10}}(0)\) and (65) we know that for \(\gamma _j(\cdot ,t)\), \(t\in I_{j,\epsilon }\), we have:

$$\begin{aligned} \sup _{s\in {\bar{B}}^k}E_j^t(s)=E_j^t(m_j(t))\le E_j^t(0)+\frac{c}{20}. \end{aligned}$$
(69)

So, to prove (68), it suffices to show the existence of \(T_j\) such that

$$\begin{aligned} |v-m_j(t)|\ge \kappa \implies E_j^t(\phi _j^t(v,T_j))<\sup _{s\in {\bar{B}}^k}E_j^t(s)-\frac{c}{5}. \end{aligned}$$

We argue by contradiction and assume that there exists a constant \(\frac{1}{4}>\kappa >0\), a sequence \(\{t_l\}_{l\in {\mathbb {N}}}\subset I_{j,\epsilon }\), and \(\{s_l\}_{l\in {\mathbb {N}}}\subset {\bar{B}}^k\) with \(|s_l-m_j(t_l)|\ge \kappa \) such that

$$\begin{aligned} E_j^{t_l}(\phi _j^{t_l}(s_l,l))\ge E_j^{t_l}(0)-\frac{c}{10}. \end{aligned}$$
(70)

Combining (70) with (69) we have \(E_j^{t_l}(\phi _j^{t_l}(s_l,l))\ge E_j^{t_l}(m_j(t_l))-\frac{c}{5}\). Since \(\phi _j^t(\cdot ,\cdot )\) is an energy decreasing flow, we have

$$\begin{aligned} E_j^{t_l}(\phi _j^{t_l}(s_l,x))\ge E_j^{t_l}(\phi _j^{t_l}(s_l,l))\ge E_j^{t_l}(m_j(t_l))-\frac{c}{5},\,\forall 0\le x\le l. \end{aligned}$$

Since both \(I_{j,\epsilon }\) and \({\bar{B}}^k\) are compact, we obtain subsequential limits \(t\in I_{j,\epsilon }\) and \(s\in {\bar{B}}^k\) with

$$\begin{aligned} E_j^t(\phi _j^t(s,x))\ge \sup _{|v|\le 1}E_j^t(v)-\frac{c_0}{5},\quad \forall x\ge 0. \end{aligned}$$
(71)

Since \(\gamma _j(\cdot ,t)\mapsto m_j(t)\) is a continuous map, \(|s_l-m_j(t_l)|\ge \kappa \) implies \(|s-m_j(t)|\ge \kappa \) . Thus we have \(\lim _{x\rightarrow \infty }|\phi _j^t(s,x)|=1\) and thus we deduce from Eq. (71) that

$$\begin{aligned} \sup _{|v|=1}E_j^t(v)\ge \sup _{|v|\le 1}E_j^t(v)-\frac{c_0}{5}. \end{aligned}$$
(72)

On the other hand, \(m_j(t)\in B^k_\frac{c}{\sqrt{10}}(0)\) implies \(|v-m_j(t)|>2/3\) for all \(v\in {\bar{B}}^k\) with \(|v|=1\). Hence, by Eq. 65 we have

$$\begin{aligned} \sup _{|v|=1}E_j^t(v)\le \sup _{|v|\le 1}E_j^t(v)-\frac{c_0}{2}\Big (\frac{2}{3}\Big )^2<\sup _{|v|\le 1}E_j^t(v)-\frac{c_0}{5}, \end{aligned}$$

which gives us the desired contradiction. \(\square \)

We define a continuous homotopy:

$$\begin{aligned} H'_{j}:I_{j,\epsilon }\times [0,1]\longrightarrow B^k_{1/2^j}(0), \end{aligned}$$

so that

$$\begin{aligned} H'_j(t,0)=0,\,\text {and}\,\inf _{t\in I_{j,\epsilon }}|H'_j(t,1)-m_j(t)|\ge \kappa _j>0. \end{aligned}$$

We are able to define \(H_j'\) due to the assumption \(\sum _i \text {Index}(u_i)=k\ge 2\). So we can choose a continuous path in \(B^k_{1/2^j}(0)\) away from the curve of \(m_j(t),\,t\in I_{j,\epsilon }.\) By Claim 4.2, there exists \(T_j\) for \(t\in I_{j,\epsilon }\) such that:

$$\begin{aligned} E^t_j\big (\phi _j^t(H_j'(t,1),T_j)\big )<E_j^t(0)-\frac{c_0}{10}. \end{aligned}$$

Let \(c_j:[0,1]\longrightarrow [0,1]\) be a cutoff function which is supported in \(I_{j,\epsilon }\), and has value one in \(I_{j,\epsilon /2}\), value zero in \([0,1]\setminus I_{j,\epsilon }.\) Define:

$$\begin{aligned} H_j(t,x)=H'_j(t,c_j(t)x), \end{aligned}$$

and

$$\begin{aligned} H_j(t,x)=0\quad \forall t\in [0,1]\setminus I_{j,\epsilon }. \end{aligned}$$

We now set \(s_j(t)=(s_j^1(t),\ldots ,s_j^k(t))\in {\bar{B}}^k\) to be

$$\begin{aligned} s_j(t)=\phi _j^t(H_j(t,1),c_j(t)T_j),\quad \text {if }t\in I_{j,\epsilon }, \end{aligned}$$

and

$$\begin{aligned} s_j(t)=0,\quad \text {if }t\in [0,1]\setminus I_{j,\epsilon }. \end{aligned}$$

We define \(\gamma '_{j}(\cdot ,t)\) to be:

$$\begin{aligned} \gamma '_{j}(\cdot ,t):=\Pi \circ \Big (\gamma _{j}(\cdot ,t)+\sum _{l=1}^k s_j^l(t)X_l(t)\Big ). \end{aligned}$$

Since \(s_j\) is homotopic to the zero map in \({\bar{B}}^k\), so \(\gamma '_{j}(\cdot ,t)\) is homotopic to \(\gamma _{j}(\cdot ,t).\)

Claim 4.3

\(\{\gamma _j'(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is a minimizing sequence.

Proof

From the energy non-increasing property of \(\{\phi _j^t(\cdot ,x)\}\in \text {Diff}({\bar{B}}^k)\) we have that for all \(t\in [0,1]\)

$$\begin{aligned} E(\gamma '_j(\cdot ,t))=E_j^t(\phi _j^t(H_j(t,1),c_j(t)T_j))\le E_j^t(H_j(t,1)). \end{aligned}$$
(73)

Moreover, from (5), we know that there exists a continuous function \(\Psi :[0,\infty )\rightarrow [0,\infty )\) with \(\Psi (0)=0\) such that

$$\begin{aligned} \Big |E(\gamma _{j,s}(\cdot ,t))-E(\gamma _j(\cdot ,t))\Big |\le \Psi \Big (\Vert \sum _{l=1}^ks_lX_l\Vert _{W^{1,2}}\Big ), \end{aligned}$$

and \(H_j(t,1)\in B^k_{1/2^j}\) implies that

$$\begin{aligned} \begin{aligned} E(\gamma '_j(\cdot ,t))&=E_j^t(\phi _j^t(H_j(t,1),c_j(t)T_j))\\&\le E_j^t(H_j(t,1))\\&\le E(\gamma _j(\cdot ,t))+\Psi \Big (\frac{1}{2^{j}}\sum _{l=1}^k\Vert X_l\Vert _{W^{1,2}}\Big ). \end{aligned} \end{aligned}$$
(74)

By (74) and that \(\gamma _j'(\cdot ,t)\) is homotopic to \(\gamma _j(\cdot ,t)\) we have that

$$\begin{aligned} W\le \lim _{j\rightarrow \infty }\max _{t\in [0,1]}E(\gamma _j'(\cdot ,t))=\lim _{j\rightarrow \infty }\max _{t\in [0,1]}E(\gamma _j(\cdot ,t))=W, \end{aligned}$$

which finishes the proof \(\square \)

Claim 4.4

There exists \(j_k'\in {\mathbb {N}}\) such that

$$\begin{aligned} d_B(\gamma '_j(\cdot ,t),\{k^l_i\}_{i=0}^{m_l})>\epsilon _l\quad \text {for }l=1,\ldots ,N_k,\forall t\in [0,1], \end{aligned}$$

for all \(j>j_k\).

Proof

The claim follows from the Assumption 66. \(\square \)

Claim 4.5

there exists \(\epsilon _J>0\) and \(J\in {\mathbb {N}}\) such that

$$\begin{aligned} d_B(\gamma _j'(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _J,\quad \forall t\in [0,1], \end{aligned}$$

for all \(j>J\).

Proof

There are three cases to consider.

Case 1 \(t\in [0,1]\setminus I_{j,\epsilon }.\)

\(\gamma _j'(\cdot ,t)=\gamma _j(\cdot ,t)\) for all j, so there exists \(\epsilon _1>0\) such that \(d_B(\gamma '_j(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _1\) for all \(j\in {\mathbb {N}}\).

Case 2 \(t\in I_{j,\epsilon /2}.\)

By Claim 4.2 we have

$$\begin{aligned} E(\gamma '_j(\cdot ,t))&=E_j^t(\phi _j^t(H_j'(t,1),T_j))\\&<E_j^t(0)-\frac{c}{10}\\&=E(\gamma _j(\cdot ,t))-\frac{c}{10},\quad \forall j\in {\mathbb {N}}, \end{aligned}$$

so

$$\begin{aligned} \lim _{j\rightarrow \infty }\max _{t\in I_{j,\epsilon /2}}E(\gamma '_j(\cdot ,t))< \lim _{j\rightarrow \infty }\max _{t\in [0,1]}E(\gamma _j(\cdot ,t))-\frac{c}{10}=W-\frac{c}{10}. \end{aligned}$$

It implies that there exists \(\epsilon _2>0\) so \(d_B(\gamma '_j(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _2\), or else by Remark 2.12\(\lim _{j\rightarrow \infty }\max _{t\in I_{j,\epsilon /2}}E(\gamma '_j(\cdot ,t))=\sum _{i=0}^nE(u_i)=W\).

Case 3 \(t\in I_{j,\epsilon }\setminus I_{j,\epsilon /2}.\)

By (74) we have

$$\begin{aligned} E(\gamma '_j(\cdot ,t))\le E(\gamma _j(\cdot ,t))+\Psi \Big (\frac{1}{2^{j}}\sum _{l=1}^k\Vert X_l\Vert _{W^{1,2}}\Big ), \end{aligned}$$

for a continuous function \(\Psi :[0,\infty )\rightarrow [0,\infty )\) with \(\Psi (0)=0\). And Lemma 3.4 implies that there exists \(\epsilon _3>0\) so \(d_B(\gamma '_j(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _3\) for j sufficiently large.

Let \(\epsilon _J=\min \{\epsilon _1,\epsilon _2,\epsilon _3\}\), so we have

$$\begin{aligned} d_B(\gamma _j(\cdot ,t),\{u_i\}_{i=0}^n)>\epsilon _J,\,\forall t\in [0,1], \end{aligned}$$

for j sufficiently large. \(\square \)

We have proved Theorem 4.1. \(\square \)

4.1 Proof of Theorem 1.1

Theorem 1.1

Let (M, g) be a closed Riemannian manifold of dimension at least three, g generic, and a non-trivial homotopy group \(\pi _3(M)\), let W be the width associated to the homotopy class \(\Omega _\beta \) (see Definition 2.15, (23)). Then there exists a collection of finitely many harmonic spheres \(\{u_i\}_{i=0}^m,\, u_i:S^2\rightarrow M\), which satisfies the following properties:

  1. (1)

    \(\sum _{i=0}^m E(u_i)=W,\)

  2. (2)

    \(\sum _{i=0}^m Index(u_i)\le 1.\)

Proof

Denote by \({\mathcal {U}}\) the collections of equivalent classes of harmonic spheres \(\{[u_i]\}_{i=0}^n\) with \(\sum _{i=0}^n\text {Index}(u_i)>1\) and \(\sum _{i=0}^nE(u_i)=W\). By Proposition B.24, \({\mathcal {U}}\) is countable and thus we can write \({\mathcal {U}}=\Big \{\{[u^1_i]\}_{i=0}^{n_1},\{[u^2_i]\}_{i=0}^{n_2},...\Big \}\) with \(\sum _{i=0}^{n_l}E(u^l_i)=W\) and \(\sum _{i=0}^{n_l}\text {Index}(u_i^l)>1\) for each \(l\in {\mathbb {N}}\).

Given a minimizing sequence \(\{\gamma _j(\cdot ,t)\}_{j\in {\mathbb {N}}}\), we consider the collection of harmonic spheres: \(\{[u_i^1]\}_{i=0}^{n_1}\), and by Theorem 4.1 there exists \(\{\gamma ^1_j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) so that

  1. (1)

    \(\gamma ^1_j(\cdot ,t)\) is homotopic to \(\gamma _j(\cdot ,t)\),

  2. (2)

    \(\{\gamma ^1_j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is a minimizing sequence,

  3. (3)

    there exists \(\epsilon _1>0\) and \(i_1\in {\mathbb {N}}\) such that

    $$\begin{aligned}d_B(\gamma _j^1(\cdot ,t),\{u_i^1\}_{i=0}^{n_1})>\epsilon _1,\quad \forall t\in [0,1],\end{aligned}$$

    for all \(j>i_1\).

We can apply Theorem 4.1 again at the minimizing sequence \(\{\gamma _j^1(\cdot ,t)\}_{j\in {\mathbb {N}}}\), with \(\{[u^2_i]\}_{i=0}^{n_2}\) the given collection of harmonic spheres, and set the compact set of harmonic spheres K to be \(K^1:=\big \{\{[u_i^1]_{i=0}^{n_1}\}\big \}\), and obtain \(\{\gamma _j^2(\cdot ,t)\}_{j\in {\mathbb {N}}}\) so that

  1. (1)

    \(\gamma ^2_j(\cdot ,t)\) is homotopic to \(\gamma _j(\cdot ,t)\),

  2. (2)

    \(\{\gamma ^2_j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is a minimizing sequence,

  3. (3)

    there exist \(\epsilon _1,\epsilon _2>0\) and \(i_1,i_2\in {\mathbb {N}}\) such that

    $$\begin{aligned}d_B(\gamma _j^l(\cdot ,t),\{u_i^l\}_{i=0}^{n_l})>\epsilon _l,\quad j>i_l,\,\forall t\in [0,1],\end{aligned}$$

    \(l=1,2\).

Proceeding inductively we can find \(\{\gamma _j^m\}_{j\in {\mathbb {N}}}\) such that

  1. (1)

    \(\gamma ^m_j(\cdot ,t)\) is homotopic to \(\gamma _j(\cdot ,t)\),

  2. (2)

    \(\{\gamma ^m_j(\cdot ,t)\}_{j\in {\mathbb {N}}}\) is a minimizing sequence,

  3. (3)

    there exist \(\epsilon _l>0\) and \(i_l\in {\mathbb {N}}\), \(l=1,\ldots ,m\), such that

    $$\begin{aligned}d_B(\gamma _j^l(\cdot ,t),\{u_i^l\}_{i=0}^{n_l})>\epsilon _l,\quad j>i_l,\,\forall t\in [0,1].\end{aligned}$$

We can choose an increasing sequence \(p_m>i_m\) such that

$$\begin{aligned} \max _{t\in [0,1]}E(\gamma ^m_{p_m}(\cdot ,t))\le W+\frac{1}{m}. \end{aligned}$$

The sequence \(\{\gamma ^m_{p_m}(\cdot ,t)\}_{m\in {\mathbb {N}}}\) is a minimizing sequence; thus, by Theorem 2.21, there exists a sequence \(\{t_m\}_{m\in {\mathbb {N}}}\subset [0,1]\) and a collection of finitely many harmonic spheres \(\{v_i\}_{i=0}^m\) with \(\sum _{i=0}^mE(v_i)=W\), such that \(\{\gamma ^m_{p_m}(\cdot ,t_m)\}_{m\in {\mathbb {N}}}\) bubble converges up to subsequence. i.e.,

$$\begin{aligned} d_B(\gamma ^m_{p_m}(\cdot ,t_m),\{v_i\}_{i=0}^m)\rightarrow 0,\,m\rightarrow \infty . \end{aligned}$$

Since \(\gamma ^m_{p_m}(\cdot ,t)\) is away from \({\mathcal {U}}\) as \(m\rightarrow \infty \), so we have \(\sum _{i=0}^m Index(v_i) \le 1\), this is what we wanted to prove. \(\square \)