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Minimal immersions of \( \mathbb{S}^2 \)

Table of Contents

The PDF version of this page can be downloaded by replacing html in the its address by pdf. For example /html/sheaf-cohomology.html should become /pdf/sheaf-cohomology.pdf.

This post is my reading note for sacks_existence_1981. I want to present the authors' ideas as clear as possible and I may probably skip a few (important) details and computations.

On Sep 22, 2019 I gave a talk at I2M, Marseille about Sacks–Uhlenbeck's work, here is note of my talk.

Local storage: Manuscript in French

1 Brief view of Sacks and Uhlenbeck's strategy.

Let \( M \) and \( N \) be compact Riemannian manifolds (without boundary), \( M \) is a surface and \( N \) is isometrically embedded in \( \mathbb{R}^k \). It was showed by Eells and Sampson eells_harmonic_1964 that if \( N \) is negatively curved than any map from \( M \) to \( N \) is homotopic to a harmonic map. The idea of Sacks and Uhlenbeck in sacks_existence_1981 consists of (1) approximating the energy functional \( E \) by a family \( E_\alpha \) satisfying Palais-Smale condition, whose nontrivial critical values can be more easily proved to exist and (2) trying to prove that the critical maps \( s_\alpha \) of \( E_\alpha \) converge in \( C^1 \)-topology.

We will first review the general machinery of Morse-Palais-Smale theory and prove the existence of \( s_\alpha \). The convergence of \( s_\alpha \) in the case of surface is due to the facts that energy functional \( E \) is a conformal invariant of \( M \), in particular \( E \) is invariant by homotheties (i.e. \( E \) remains unchanged when we zoom in and out), which allows us to justify the \( C^1 \)-convergence (under conditions of \( N \)) except at finitely many points using a local estimate and a suitable covering of \( M \).

Sacks and Uhlenbeck used an extension result for harmonic map, in an elegant argument to prove that if the above sequence \( \{s_\alpha\} \) fails to converge at a point, for a certain surface \( M \), then one has a nontrivial harmonic map from \( \mathbb{S}^2 \) to \( N \). Therefore if such sequence \( \{ s_\alpha \}\) from \(\mathbb{S}^2\) to \( N \) exists, for example when \( \pi_k (N) \) is nontrivial for a certain \( k\geq 2 \) then, whether \( s_\alpha \) converges or not, there exists a nontrivial harmonic map from \( \mathbb{S}^2 \) to \( N \).

Finally, the theory of branched immersion of surfaces by Gulliver-Osserman-Royden gulliver_theory_1973 can be applied to show that the harmonic map obtained this way is a conformal, branched, minimal immersion of \( \mathbb{S}^2 \) to \( N \).

2 General machinery by Morse-Palais-Smale.

2.1 Perturbed functionals \( E_\alpha \).

Let \( s: M \longrightarrow N \hookrightarrow \mathbb{R}^k \) be a map from a compact surface \( M \) to a compact Riemannian manifold \( N \) isometrically embedded into \( \mathbb{R}^k \). Recall that the energy functional of \( s \) is given by \( E(s):= \frac{1}{2}\int_M |ds|^2 dV_M = \frac{1}{2}\int_M \langle s^* g_N, g_M \rangle dV_M \). The perturbed energy functionals are \[ E_\alpha(s) := \int_M\left(1 + |ds|^2\right)^\alpha dV,\quad \alpha \geq 1 \] We will suppose, by rescaling the metric \( g_M \) of \( M \) that the volume of \( M \) is 1, so when \( \alpha=1 \), \( E_1 = 1+ 2E(s) \) is just the previously defined energy. Using \( (a+b)^\alpha \geq a^\alpha + b^\alpha \) and Jensen's inequality, one has \( E_\alpha(s) \geq 1 + (2E(s))^\alpha \) for all \( \alpha \geq 1 \). Also, since we only interest in the case \( \alpha \) close to \( 1 \), let us also suppose that \( \alpha \) from now on is smaller than 2.

By Sobolev embedding, one has \( W^{1,2\alpha}(M, \mathbb{R}^k) \subset C^0(M, \mathbb{R}^k) \) compactly for all \( \alpha >1 \). It then makes sense to talk about \( W^{1,2\alpha}(M,N) \subset C^0(M,N) \) which consist of elements of \( W^{1,2\alpha}(M, \mathbb{R}^k)\subset C^0(M, \mathbb{R}^k) \) whose image lies in \( N \).

The spaces \( C^\infty (M,N)\subset W^{1,2\alpha}(M,N)\subset C^0(M,N)\), where \( \alpha>1 \), are of the same homotopy type and the inclusions are homotopy equivalences. In particular, their connected components are naturally in bijection.

We will also need a version of Morse theory for Banach manifolds, also developed by R. Palais in palais_lusternik-schnirelman_1966. For the terminologies, in the same way that a manifold is modeled by\( \mathbb{R}^n \), a Banach manifold is modeled by Banach spaces. A Finsler manifold is a Banach manifold with a norm on its tangent space that is comparable with the norm of Banach charts.

  1. If \( F \) is a \( C^2 \) functional on a complete \( C^2 \) Finsler manifold \( L \), \( F \) is bounded below and \( F \) satisfies Palais-Smale condition (C) then
    1. The functional \( F \) admits minimum on each connected component of \( L \).
    2. If \( F \) has no critical value in \( [a,b] \) then the sublevel \( \{F\leq b\} \) retracts by deformation to the sublevel \( \{F\leq a\} \).
  2. The pair \( (L,F)=(W^{1,2\alpha}(M,N), E_\alpha) \) with \( \alpha > 1 \) satisfies the condition of the first part.

The Palais-Smale condition is as follows:

(C): Let \( S\subset L \) be a subset on which \( |F| \) is bounded, but \( |dF| \) is not bounded away from \( 0 \). Then there exists a critical point of \( F \) in \( \bar S \).

The strategy to prove Theorem thm:Palais-2 is, as in finite dimensional case, to use a pseudo-gradient flow of \( F \) whose existence is due to a partition of unity of \( L \) (instead of a Riemannian metric on \( L \)). The role of Palais-Smale condition in the proof is as follows: Suppose that \( \{x_n\} \) is a sequence in a connected component \( L_1 \) of \( L \) such that \( F(x_n) \) tends to \( \inf_{L_1} F \), then using the pseudo-gradient flow of \( F \), we can suppose that \( |dF(x_n)| \) is arbitrarily small, in particular, we can suppose that \( |dF(x_n)|\to 0 \). Choose a sequence \( \{y_n\} \) of regular points near \( x_n \) such that \( F(y_n) \to \inf_{L_1} F \) and \( |dF(y_n)|\to 0\) and use (C) for \( S=\{y_n\} \), one obtains a limit point \( y_\infty \) of \( \{y_n\} \), hence also of \( \{x_n\} \), which minimises \( F \).

As a consequence of Theorem thm:Palais-2, one has:

The minimum of \( E_\alpha \) in each connected component \( C \) of \( W^{1,2\alpha}(M,N) \), \( \alpha >1 \) is taken by some \( s_\alpha \in C^\infty(M,N) \) and there exists \( B>0 \) depending on the component \( C \) such that \[ \min_{C}E_\alpha \leq (1+B^2)^\alpha \]

By Theorem thm:Palais-2, \( E_\alpha \) admits minimum at \( s_\alpha \) on each component \( C \) of \( W^{1,2\alpha}(M,N) \). By writing down the Euler-Lagrange equation of \( E_\alpha \) and apply regularity estimates, one can prove that \( s_\alpha \) is actually smooth. By Theorem thm:Palais-1, the preimage of \( C \) by inclusion \( C^\infty(M,N)\subset W^{1,2\alpha}(M,N) \) is a connected component \( C' \) of \( C^\infty(M,N) \) over which \( s_\alpha \) is the minimum of \( E_\alpha \). Take \( B = \sup_M |du| \) for an arbitrary element \( u\in C' \) and the conclusion follows.

Corollary cor:Palais-1-2 is trivialised when \( W^{1,2\alpha}(M,N) \) is connected (for one \( \alpha \) or equivalently for all \( \alpha \)). In this case, \( s_\alpha \) is a constant map and \( B=0 \).

To establish a nontrivial analog of Corollary cor:Palais-1-2 in the case where the spaces of maps from \( M \) to \( N \) are connected, we will have to look at the submanifold \( N_0\cong N \) formed by constant maps.

2.2 Tubular neighborhood of the submanifold of trivial maps.

Fix \( y\in N \), considered as a constant maps in \( N_0 \). We will summarise a few facts about the tangent space of \( W^{1,2\alpha}(M,N) \) at \( y \) in the following Remark.

These facts come from the differential structure of the Banach manifold \( W^{1,2\alpha}(M,N) \) that so far has not been introduced, since we only consider \( W^{1,2\alpha}(M,N) \) as a closed subset of \( W^{1,2\alpha}(M,\mathbb{R})^{\oplus k} \) (so only a topological structure was given). We summarise here, and refer to palais_foundations_1968, how a differential structure is given to \( W^{k,p}(M,N) \) with \( k,p \) such that \( W^{k,p}(M) \hookrightarrow C^0(M) \):

  • Let \( \xi \) be a finite dimensional vector bundle over a compact manifold \( M \), then \( W^{k,p}(\xi,M) \) can be defined as the Banach space of sections of \(\xi \) that are locally \( W^{k,p} \). A norm of \( W^{k,p}(\xi,M) \) can be given using a metric of \( \xi \) and a volume form of \( M \), but by compactness of \( M \), its equivalent class is independent of such choices.
  • Let \( E \) be a fiber bundle over \( M \), in our case, \( E = N\times M \), and \( s \in C^0(E) \) be a continuous section. It can be proved that there exists an open subset \( \xi \) of \( E \) containing \( s \) such that \( \xi \to M \) has a vector bundle structure. We say that \( s\in W^{k,p}(E,M) \) if \( s\in W^{k,p}(\xi,M) \) and it turns out that this definition is independent of the choice of \( \xi \). This defines \( W^{k,p}(E,M) \) set-theoretically.
  • The differential structure of \( W^{k,p}(E,M) \) is given by the atlas \( W^{k,p}(\xi,M) \).
  1. The tangent \( T_y W^{1, 2\alpha}(M,N) \) can be identified with \( W^{1,2\alpha}(M, T_y N)\). The subspace \( T_y N_0 \) contains constant maps from \( M \) to \( T_y N \).
  2. The fiber \( \mathcal{N}_y \) over \( y \) of the normal bundle \( \mathcal{N} \) of \( N_0 \) can be identified with \[ \mathcal{N}_y = \left\{v\in W^{1,\alpha}(M, T_y N): \int_M v dV = 0\right\} \]

The exponential map on \( TW^{1,2\alpha}(M,N) \) can be defined as follows:

\begin{align*} e:\ TW^{1,2\alpha}(M,N) & \longrightarrow W^{1,2\alpha}(M,N)\\ (s,v) &\longmapsto \left(x\mapsto \exp_{s(x)}v(x) \right) \end{align*}

where \( s\in W^{1,2\alpha}(M,N) \) and \( v\in T_s W^{1,2\alpha}(M,N) \) is a \( W^{1,2\alpha} \) vector field along \( s(x) \). With the representation of normal bundle \( \mathcal{N} \) as Remark rem:tangent-palais, the restriction of \( e \) on \( \mathcal{N} \) is given by

\begin{align*} \restr{e}{\mathcal{N}}:\ \mathcal{N} &\longrightarrow W^{1,2\alpha}(M,N)\\ (y,v) &\longmapsto \left(x\mapsto \exp_y (v(x)) \right) \end{align*}

where \( y\in N_0 \cong N \) and \( v \in W^{1,2\alpha}(M, T_yN) \).

The restriction \( \restr{e}{\mathcal{N}} \) of \( e \) on \( \mathcal{N} \) is a local diffeomorphism mapping a neighborhood of the zero-section of \( \mathcal{N} \) onto a neighborhood of \( N_0 \) in \( W^{1,2\alpha}(M,N) \).

It can be calculated that \[ de_{(y,0)}(a,v) = \left(x\mapsto a + v(x)\right) \in T_yW^{1,2\alpha}(M,N) = W^{1,2\alpha}(M, T_yN) \] for \( a\in T_y N \) and \( v\in \mathcal{N}_y \subset W^{1,2\alpha}(M, T_y N) \). It is invertible since \( a \) is tangential to \( N_0 \) and \( v\in \mathcal{N}_y \) is in the normal component. The Inverse function theorem applies.

2.3 Critical values of \( E_\alpha \).

The exponential map previously defined on the normal bundle of \( N_0 \) in \( W^{1, 2\alpha}(M,N) \) allows us to retract by deformation a small neighborhood of \( N_0 \) to \( N_0 \). We will prove that if the energy \( E_\alpha(s) \) is sufficiently close to \( 1=E_\alpha(N_0) \) then \( s \) is sufficiently \( W^{1,2\alpha} \)-close to \( N_0 \) and hence can be retracted to \( N_0 \), in other words, \( E_\alpha^{-1}[1, 1+\delta] \) retracts by deformation to \( N_0 = E_\alpha^{-1}(1) \).

Given \( \alpha>1 \), there exists \( \delta >0 \) depending on \( \alpha \) such that \( E_\alpha^{-1}[1, 1+\delta] \) retracts by deformation to \( E_\alpha^{-1}(1) = N_0 \).

Let \( s\in E_\alpha^{-1}[1, 1+\delta] \), using \( (a+b)^\alpha \geq a^\alpha + b^\alpha \), one has \[ 1 + \delta > \int_M (1+|ds|^2)^\alpha dV > 1 + \int_M |ds|^{2\alpha}dV \] therefore \( \|ds\|_{L^{2\alpha}}\leq \delta ^{1/{2\alpha}} \). By Poincaré-Wirtinger inequality, \( \|s-\int_M s\|_{W^{1,2\alpha}}\leq C \delta^{1/4} \) where \( C \) is the Poincaré-Wirtinger constant.

By Sobolev embedding, \( \max_M |s-\int_M s| \leq C_\alpha \|s -\int_M s\|_{W^{1,2\alpha}} \) where the Sobolev constant \( C_\alpha \) can no longer be chosen uniformly in \( \alpha \to 1 \). Fix an \( x_0\in M \), one has \[ d_{W^{1,2\alpha}}(s, N_0) \leq \|s - s(x_0)\|_{W^{1,2\alpha}} \leq \left\|s-\int_M s\right\|_{W^{1,2\alpha}} + \left|\int_M s - s(x_0)\right| \leq C_\alpha \delta^{1/4} \]

Now choose \( \delta \ll 1 \) depending on \( \alpha \) such that \( s \) is in the neighborhood of \( N_0 \) given by Lemma lem:local-isom-e, \( s \) can be written as \[ s(x) = e(y,v(x)) = \exp_y v(x) \] where \( y\in N_0 \) and \( v\in W^{1,2\alpha}(M, T_yN) \) depend continuously on \( s\in W^{1,2\alpha}(M,N) \). We can define the deformation retraction by

\begin{align*} \sigma:\ E^{-1}_\alpha [1,1+\delta]\times [0,1] & \longrightarrow E^{-1}_\alpha[1,1+\delta]\\ (s,t) &\longmapsto \left( x\mapsto \exp_y tv(x)\right) \end{align*}

It is clear that \( \sigma \) is continuous and \( \sigma_0 \) is a retraction. The only thing to check is that the image of \( \sigma \) remains in \( E_\alpha^{-1}[1,1+\delta] \) at all time. This can be checked by showing that \( \frac{d}{dt}E_\alpha(\sigma_t) \geq 0\), hence \( E_\alpha(\sigma_t) \leq E_\alpha(\sigma_1) \leq 1+\delta \) for all \( 0\leq t\leq 1\).

We will now prove the existence of nontrivial critical value of \( E_\alpha \) in an interval \( (1, B) \) for a certain \( B>1 \) sufficiently big independently of \( \alpha > 1 \).

Fix \( z_0\in M \) and consider the map

\begin{align*} p: \ C^0(M,N) &\longrightarrow N\\ s &\longmapsto f(z_0) \end{align*}

then \( p \) is a fiber bundle and therefore is a Serre fibration. In fact fix \( q_0\in N \) then for all \( q\in N \) near \( q_0 \), there is a vector field \( v_q \) supported in a small ball centered at \( q_0 \) such that the flow of \( v_q \) from time 0 to 1 turns \( q_0 \) to \(q\), i.e. \( {\Phi_{v_q}}_0^1(q_0)=q \), and that \( v_q \) varies continuously in \( q \). Then any fiber \( p^{-1}(q) \) can be identified with \( p^{-1}(q_0) \) using the flow of \( v_q \). We will denote by \( \Omega(M,N) \) the topological fiber of \( p \).

We will use a few facts from algebraic topology, briefly summarised here.

  1. (Long exact sequence of homotopy) Let \( p: E \longrightarrow B \) be a fiber bundle of fiber \( F = p^{-1}(b_0) \ni f_0 \), then one has the following long exact sequence \[ \xymatrix{ \dots \ar@{->}[r]^{\partial} & \pi_n(F) \ar@{->}[r]^{\iota_*} & \pi_n(E) \ar@{->}[r]^{p_*} & \pi_n(B) \ar@{->}[r]^{\partial} & \pi_{n-1}(F) \ar@{->}[r] & \dots \ar@{->}[r] & \pi_0(E) \ar@{->}[r] & 0 } \] where \( \iota: F \longrightarrow E \) is the inclusion.
  2. If \( p \) admits a global section \( s \), then one has a retraction \( s_* \) of \( p_* \): \[ \xymatrix{ \pi_n(E) \ar@{->}[r]^{p_*} & \pi_n(B) \ar@/^/@{->}[l]^{s_*} } \] hence \( p_* \) is surjective and \( \partial \) factors through \( 0 \), which gives us the short exact sequence \[ \xymatrix{ 0 \ar@{->}[r] & \pi_n(F) \ar@{->}[r]^{\iota_*} & \pi_n(E) \ar@{->}[r]^{p_*} & \pi_n(B) \ar@/^/@{->}[l]^{s_*} \ar@{->}[r] & 0 } \] where \( p_* \) admits a retraction \( s_* \), so the short exact sequence splits and we have \[ \pi_n(E) \cong \pi_n(F) \oplus \pi_n(B). \]

Now apply this result to the fiber bundle \( p: C^0(M,N) \longrightarrow N \) of fiber \( \Omega(M,N) \), which has \( N_0 \) as a global section, one obtains \[ \pi_n(C^0(M,N)) \cong \pi_n(N) \oplus \pi_n(\Omega(M,N)). \]

If \( C^0(M,N) \) is not connected, or if \( \Omega(M,N) \) is not contractible, then there exists \( B>0 \) such that for all \( \alpha >1 \), \( E_\alpha \) has critical values in the interval \( (1, (1+B^2)^\alpha) \).

In particular, if \( M=\mathbb{S}^2 \) and if the universal covering \( \tilde N \) of \( N \) is not contractible then \( E_\alpha \) has critical values in \( (1, (1+B^2)^\alpha) \).

If \( C^0(M,N) \) is not connected, one only needs to apply Corollary cor:Palais-1-2 to a connected component of \( W^{1,2\alpha}(M,N) \) not containing \( N_0 \). We now suppose that \( C^0(M,N) \) is connected and \( \Omega(M,N) \) is not contractible.

In this case, there exists \( n>0 \) such that \( \pi_n(\Omega(M,N)) \) is nontrivial and contains a nonzero element \( \gamma:\ \mathbb{S}^n \longrightarrow \Omega(M,N) \) which is not homotopic to any \( \tilde\gamma:\ \mathbb{S}^n \longrightarrow N_0 \) in \( \pi_n(C^0(M,N)) \).

Choose \( B:= \max_{\theta\in \mathbb{S}^n,x\in M} |d\gamma(\theta)(x)| \) then by definition \[ E_\alpha(\gamma(\theta)) \leq (1+B^2)^\alpha\quad\forall \theta\in \mathbb{S}^n,\alpha>1. \] If \( E_\alpha \) has no critical value in \( [1+\frac{\delta_\alpha}{2}, (1+B^2)^\alpha] \) where \( \delta_\alpha \) is given by Proposition prop:crit-val-1, then by Theorem thm:Palais-2, \( E_\alpha^{-1}[1, (1+B^2)^\alpha] \) retracts by deformation to \( E_\alpha^{-1}[1, 1+\delta_\alpha] \) which retracts by deformation to \( E_\alpha^{-1}(1)=N_0 \). But this means that \( \gamma \) is homotopic to a certain \( \tilde\gamma\in \pi_n(N) \), which is a contradiction.

As an application, if \( M= \mathbb{S}^2 \) and the universal covering \( \tilde N \) is not contractible then the long exact sequence of homotopy for the bundle \( \tilde N \longrightarrow N\) with fiber of dimension \( 0 \), gives \[ \pi_n(\tilde N) = \pi_n(N),\quad \forall n\geq 2. \] Since \( \tilde N \) is simply-connected and not contractible, there exists \( n\geq 2 \) such that \( 0\ne\pi_n(\tilde N) = \pi_n(N) = \pi_{n-2}(\Omega(\mathbb{S}^2,N)) \), where the last equality follows from Exponential law in Pointed category (the role of cartesian product is played by smash product) and the fact that smash product of spheres is a sphere. The general argument applies.

3 Local results: Estimates and extension.

We will say that the map \( s: M \longrightarrow N \) is a critical point of \( E_\alpha \) on a small disc \( D(R)\subset M \) if \( s \) satisfies the Euler-Lagrange equation of \(E_\alpha\) (as functional on \( W^{1,2\alpha}(M,N) \)) on \( D(R) \).

Rescaling \( (D(R), g_M) \), where \( R\ll 1 \) and \( g_M \) is \( \epsilon \)-close to the Euclidean metric, to the unit disc \( D \) one obtains a metric \( \tilde g_M \) that is still \( \epsilon \)-close to Euclidean metric. The curvature of \( \tilde g_M \) is \( R^2 \) times smaller than that of \( g_M \).

If \( s:\ D(R) \longrightarrow N \) is a critical map of \( E_\alpha \) on \( D(R) \), then the composition \( \tilde s \) of \( s \) and the rescaling operator \( D \longrightarrow D(R) \) satisfies the Euler-Lagrange equation of \( \tilde E_\alpha = R^{2(1-\alpha)}\int_D (R^2 + |d\tilde s|^2)^\alpha d\tilde V \) where \( d\tilde V \) is the volume form of the rescaled metric \( \tilde g_M \). We will abusively use the same notation for \( \tilde s \) and \( s \) and regard \( s \) as a map on the unit disc \( D \).

For all \( p\in (1,+\infty) \), there exists \( \epsilon>0 \) and \( \alpha_0 >1 \) depending on \( p \) such that if

  • \( s:\ (D,\tilde g) \longrightarrow N \) is a critical map of \( E_\alpha \) on \( D(R) \)
  • \( E(s) < \epsilon \), \( 1 < \alpha <\alpha_0 \)

then \[ \|ds\|_{W^{1,p}(D')} < C(p,D') \|ds\|_{L^2(D)},\quad \text{for all disc } D'\Subset D \]

In fact \( \alpha_0, \epsilon \) and \( C(p,D') \) depend on the rescaled metric \( \tilde g\) on \( D \), but if \( R \ll 1 \) and \( \tilde g \) is very close to Euclidean metric, then one can choose these parameters independently of \( \tilde g \).

A consequence of (the proof of) Lemma lem:main-est is the following global result:

There exists \( \epsilon' >0 \) and \( \alpha_0 > 1 \) such that if

  • \( s: M \longrightarrow N \) is critical map of \( E_\alpha \)
  • \( E(s)<\epsilon' \), 1 < α <α_0

then \( s\in N_0 \) and \( E(s) = 0 \).

We proved in the last section that, under certain algebraic topological condition on \( N \), \( E_\alpha \) admits critical value \( v_\alpha \in (1, (1+B^2)^\alpha) \). We now can conclude that, by Theorem thm:4-trivial-energy, the critical values \( v_\alpha \) are bounded away from \( 1 \), i.e. \( \inf_{\alpha} v_\alpha > 1 \).

We will also need the following extension theorem:

If \( s:\ D\setminus \{0\} \longrightarrow N \) is a harmonic map with finite energy \( E(s) < \infty \), then \( s \) extends to a smooth harmonic map \( \tilde s:\ D \longrightarrow N \).

4 Convergence of critical maps of \( E_{\alpha} \).

We proved in Theorem thm:3-nontrivial-crit that if \( C^0(M,N) \) is not connected or if \( \Omega(M,N) \) is not contractible, then there exists a family \( \{s_\alpha\} \) of critical maps of \( E_\alpha \) with bounded, nontrivial energy \( E_\alpha(s_\alpha) < B \). Since

  • \( \int_M |ds_\alpha|^2 \leq \left(E_\alpha(s_\alpha)-1\right)^{1/\alpha} \) is bounded uniformly on \( \alpha \)
  • \(\|s_\alpha\|_{L^\infty}\) is bounded by compactness of \( N \).

the \( W^{1,2}(M, \mathbb{R}^k) \)-norms of \( \{s_\alpha\} \) are bounded. By reflexivity of Sobolev spaces, there exists a subsequence \( \{s_\beta\} \) weakly converging to \( s \) in \( W^{1,2}(M,\mathbb{R}^k) \) with \[ \|s\|_{W^{1,2}}\leq \liminf_{\beta\to 1} \|s_\beta\|_{W^{1,2}} \] We do not know at this moment if the convergence is \( C^0 \), or if \( s \) is continuous, or even if the image of \( s \) remains in \( N \). The following key lemma answer these questions on a small disc of \( M \) in the case the energy of \( s_\alpha \) is small.

There exists an \( \epsilon>0 \), in fact given by the Main estimate Lemma lem:main-est with \( p=4 \), such that if

  • \( s_\alpha:\ D(R) \longrightarrow N\subset \mathbb{R}^k \) are critical maps of \( E_\alpha \) in \( W^{1, 2\alpha}(D(R),N) \),
  • \( E(s_\alpha) < \epsilon \) and \( s_\alpha \) converges weakly to \( s \) in \( W^{1,2}(D(R),\mathbb{R}^k) \),

then

  • the restriction of \( s \) on \( \overline{D(R/2)} \) is smooth harmonic map with image in \( N \),
  • \( s_\alpha \to s \) in \( C^1(\overline{D(R/2)},N)) \).

There are two different ways to define convergence of a sequence \( s_n \) to \( s \) in \( C^1(\Omega) \) on an open set \( \Omega \):

  1. The sequence \( s_\alpha \) and \( s \) extend to \( C^1(\bar\Omega) \) and have finite norm \( \max_\Omega |s| + \max_\Omega|ds|\) and \(\max_\Omega |s_\alpha| + \max_\Omega|ds_\alpha|\) and \[ \max_\Omega |s_\alpha - s| + \max_\Omega |ds-ds_\alpha| \to 0. \] In this case, we will say that \( s_\alpha \) converges to \( s \) in \( C^1(\bar\Omega) \).
  2. \( C^1(\Omega) \) is topologised by a family of seminorms \( \Gamma_K:\ s\longmapsto \max_{K} |s| + \max_{K}|ds| \) for \( K\Subset\Omega \). This makes \( C^1(\Omega) \) a Fréchet topological vector space. If the sequence \( s_\alpha \) converges to \( s \) under this topology then we will say that \( s_\alpha \) converges uniformly to \( s \) on compacts of \( \Omega \).

We consider \( s_\alpha \) and \( s \) as maps from the unit disc \( D \) to \( \mathbb{R}^k \), then by Main estimate Lemma lem:main-est for \( p=4 \), since \( E(s_\alpha) < \epsilon \), one has: \[ \|ds_\alpha\|_{W^{1,4}(D(1/2), \mathbb{R}^k)} \leq C(4, D(1/2)) \| ds_\alpha\|_{L^2(D)} = C(4, D(1/2)) E(s_\alpha)^{1/2} \] So \( \{s_\alpha\} \) is bounded in \( W^{2,4}(D(1/2), \mathbb{R}^k) \) which is embedded compactly into \( C^1(\overline{D(1/2)}, \mathbb{R}^k) \).

We now can prove that \( s_\alpha \) converges strongly to \( s \) in \( C^1(\overline{D(1/2)},\mathbb{R}^k) \): If there was a subsequence \( \{s_\beta\} \) whose restriction to \( \overline{D(1/2)} \) remains \( C^1 \)-away from \( s \), then by compactness of \( W^{1,4}(D(1/2), \mathbb{R}^k) \hookrightarrow C^1(\overline{D(1/2)}, \mathbb{R}^k) \), we can suppose that \( \{s_\beta\} \) converges in \( C^1 \) to a certain \( \bar s\ne s \) on \( \overline{D(1/2)} \). But as a subsequence of \( \{s_\alpha\} \), \( \{s_\beta\} \) converges weakly to \( s \) on \( D \), hence on \( \overline{D(1/2)} \), we than obtain a contradiction using the uniqueness of weak limit.

By considering the Euler-Lagrange equation and letting \( \alpha\to 0 \), one concludes that \( s \) is a harmonic map from \( D(1/2) \) to \( N \).

The global convergence of \( \{s_\alpha\} \) can be established by a well-chosen covering of \( M \) by small balls or radius \( R \).

Let \( s_\alpha:M \longrightarrow N\subset \mathbb{R}^k \) be critical maps of \( E_\alpha \) on \( M \) such that \( s_\alpha \) converges weakly to \( s \) in \( W^{1,2}(M, \mathbb{R}^k) \) and \( E(s_\alpha) < B \). Then there exists \( l=l(B,N) \) such that given any \( m>0 \), one can find a sequence \( \{x_{m,1},\dots, x_{m,l}\}\subset M \) and a subsequence \( \{s_{\alpha(m)}\} \) of \( \{s_\alpha\} \) such that \[ s_{\alpha(m)} \longrightarrow s\ \text{in } C^1\left(M\setminus\bigcup_{i=1}^{l} D(x_{m,i}, 2^{-m+1}),N\right) \]

We cover \( M \) by finitely many balls \( D(y_i, 2^{-m}) \) such that each point is covered at most \( h \) times by the bigger balls \( D(y_i,2^{-m+1}) \). By Lemma lem:uni-loc-finite-cover, \( h \) can be chosen independently of \( m \) as \( 2^{-m} \to 0 \).

Since \( \sum_i \int_{D(y_i, 2^{-m+1})} |ds_\alpha|^2 < B h\), choosing \( l = \lceil \frac{Bh}{2\epsilon} \rceil \), we see that there are at most \( l \) balls \( D(y_{\alpha,i}, 2^{-m+1}) \) with centers depending on \( \alpha \), on which the energy \( E(s_\alpha) \) is more than \( \epsilon \). Passing to a subsequence \( \{s_{\alpha(m)}\} \) of \( \{s_\alpha\} \), we can suppose that \( \{y_{\alpha(m),i} \}\) converges to \( x_{m,i} \) as \( \{\alpha(m)\} \to 1 \). But since the points \( \{y_i\} \) are of finite number and separated, \( y_{\alpha(m),i}\equiv x_{m,i} \) eventually and we can suppose that the bad balls \( D(y_{\alpha(m),i}) \) where energy of \( s_{\alpha(m)} \) surpasses \( \epsilon \) are the same for every \( s_{\alpha(m)} \).

Now apply Lemma lem:key-sacks-uhlenbeck to the sequence \( \{s_{\alpha(m)}\} \) on all the other \( 2^{-m+1} \)-balls, one sees that \( \{s_{\alpha(m)}\} \) converges in \( C^1 \) to \( s \) on all \( \overline{D(y_i, 2^{-m})} \) except those centered at \( x_{m,i} \). The conclusion follows.

Using a diagonal argument, we can find a subsequence \( \{s_\beta\} \) of \( \{s_\alpha\} \) that converges to \( s \) uniformly on compacts of \( M\setminus \{x_1,\dots, x_l\} \).

Let \( s_\alpha:M \longrightarrow N\subset \mathbb{R}^k \) be critical maps of \( E_\alpha \) on \( M \) such that \( s_\alpha \) converges weakly to \( s \) in \( W^{1,2}(M, \mathbb{R}^k) \) and \( E(s_\alpha) < B \). Then there exist at most \( l \) points \( x_1,\dots, x_l \) in \( M \), where \( l \) is given by Proposition prop:2-sacks-uhlenbeck, and a subsequence \( \{s_\beta\} \) of \(\{s_\alpha\}\) such that \[ s_\beta \longrightarrow s \ \text{in } C^1(M\setminus\{x_1,\dots, x_l\}, \mathbb{R}^k) \ \text{uniformly on compacts}. \]

By passing to a subsequence \( \{m_k\} \) of \(\{ m\} \), we can suppose that \(\{x_{m,i}\}\) converges to \( x_i \) in \( M \). Choose the diagonal subsequence \( \{s_\beta\} \) from \( \{s_{\alpha(m)}\} \) that consists of \( s_{\alpha(m)(a_m)} \) where \( a_m \) is sufficiently big such that \( \alpha(m)(a_m) \) is increasing and \( \|s_{\alpha(m)(b)} - s_{\alpha(m)(c)} \|_{C^1(M\setminus \cup_i D(x_{m,i},2^{-m+1})} < \frac{1}{m} \) for all \( b,c\geq a_m \). Then the sequence \( \{s_\beta\} \) converges uniformly on compacts of \( M\setminus\{x_1,\dots,x_l\} \) because \( \{\bigcup_i D(x_{m,i}, 2^{-m+1})\}_m \) is an exhaustive family of compacts of \( M\setminus\{x_1,\dots,x_l\} \).

With the same notation as Theorem thm:5-convergence-crit-map,

  1. The image \( s(M\setminus\{x_1,\dots,x_l\})\) lies in \( N \). Also, using the Euler-Lagrange equation, one sees that \( s \) is a (smooth) harmonic map from \( M\setminus\{x_1,\dots, x_l\} \) to \( N \).
  2. Since \( E(s) \leq \|s\|_{W^{1,2}}^2 \leq \liminf_{\alpha\to 1} \|s_\alpha\|^2 <+\infty \), \( \restr{s}{M\setminus\{x_1,\dots, x_l\}} \) extends to a harmonic map \( \tilde s:\ M \longrightarrow N \). We can therefore suppose that the limit \( s \) of Theorem thm:5-convergence-crit-map is smooth harmonic map on \( M \) and of image in \( N \).

5 Nontrivial harmonic maps from \( \mathbb{S}^2 \).

We will now prove the existence of nontrivial harmonic maps from \( \mathbb{S}^2 \) to a compact Riemannian manifold \( N \) satisfying the conditions of Theorem thm:3-nontrivial-crit.

The following theorem does not suppose any condition on \( N \).

Let \( M \) be a compact surface and \( s_\alpha \) be critical maps of \( E_\alpha \). Suppose that

  • \( s_\alpha \) converges in \( C^1 \) to \( s \) uniformly on compacts of \( M\setminus\{x_1,\dots,x_l\} \) but not on \( M\setminus\{x_2,\dots,x_l\} \).
  • \( E(s_\alpha) < B \)

Then there exists a nontrivial harmonic map \( s_*: \mathbb{S}^2 \longrightarrow N \).

Before proving the theorem, let us state its corollary.

If the universal covering \( \tilde N \) of \( N \) is not contractible then there exists a nontrivial harmonic map \( s: \mathbb{S}^2 \longrightarrow N \).

By Theorem thm:3-nontrivial-crit and Theorem thm:4-trivial-energy, there exist critical maps \( s_\alpha: \mathbb{S}^2 \longrightarrow N \) of \( E_\alpha \) corresponding to critical values \( E_\alpha(s_\alpha) \) in \( (1+\delta,B) \). We claim that \( \{s_\alpha\} \) cannot converge in \( C^1(M) \) to a trivial harmonic map \( s\in N_0 \). In fact, if it did, \[ 1+\delta \leq \lim_{\alpha\to 1} \int_M (1+|ds_\alpha|^2)^\alpha dV = \int_M (1+|ds|^2)dV = 1 \] which is contradictory.

Therefore, we only have two possibilities:

  • \( \{s_\alpha\} \) does not converge in \( C^1(M) \) to \( s \), then by Theorem thm:6-final-sacks-uhlenbeck, there exists a nontrivial harmonic map \( s_*: \mathbb{S}^2 \longrightarrow N \).
  • If \( \{s_\alpha\} \) converges in \( C^1(M) \) to a certain \( \tilde s \), then as argued above, \( \tilde s \) is nontrivial.

In both cases, nontrivial harmonic map from \( \mathbb{S}^2 \) to \( N \) exists.

Let us now prove Theorem thm:6-final-sacks-uhlenbeck.

If there is no \( C^1 \) convergence near \( x_1 \), we claim that:

For all \( C>0 \) and \( \delta >0 \), there exists \( \alpha >1 \) arbitrarily close to 1 such that \[ \max_{\overline{D}(x_1,2\delta)}|ds_\alpha| > C. \] Moreover, we can suppose that \( \max_{\overline{D}(x_1,2\delta)}|ds_\alpha| = \max_{D(x_1,\delta)}|ds_\alpha| \).

Suppose that was not the case, then there exist \( C,\delta >0 \) such that \( \max_{D(x_1,2\delta)} |ds_\alpha| \leq C \) for all \( \alpha>1 \) sufficiently close to 1. Choose a radius \( R \ll \delta \) such that \[ \int_{D(x_1,R)} |ds_\alpha|^2 \leq \pi R^2 C^2 <\epsilon \] It suffices to apply Key lemma lem:key-sacks-uhlenbeck to see that \( s_\alpha \to s \) in \( C^1(D(x_1,R/2)) \), hence \( s_\alpha \) converges to \( s \) in \( C^1(M\setminus\{x_2,\dots,x_l\}) \) uniformly on compacts. Moreover, since \( \{ds_\alpha\}\) converges uniformly to \( ds \) on \( \overline{D}(x_1,2\delta)\setminus D(x_1,\delta) \), we can suppose, with \( \alpha \) sufficiently close to \( 1 \), that the maximum is actually attained in \( D(x_1,\delta) \).

Therefore, we can choose a sequence \( \{C_n\} \) increasing to \( +\infty \) and \( \{\delta_n\} \) decreasing to \( 0 \), such that \( C_n\delta_n \) diverges to \( +\infty \) and there exists a sequence \( \{\alpha_n\} \) decreasing to \( 1 \) such that \[ \left| ds_{\alpha_n}(y_n)\right|:=\max_{D(x_1,\delta_n)}|ds_{\alpha_n}|=\max_{D(x_1,2\delta_n)}|ds_{\alpha_n}| = C_n \]

We define

\begin{align*} \tilde s_{\alpha_n}:\ D(\delta_n C_n) & \longrightarrow N \\ x &\longmapsto s_{\alpha_n}(y_n + C^{-1}_nx) \end{align*}

then \( |d\tilde s_{\alpha_n}(0)|= \max_{D(C_n\delta_n)}|d\tilde s_{\alpha_n}| = 1 \).

Fix any large \( R <+\infty \), since \( C_n\delta_n\to +\infty \), \( \tilde s_{\alpha_n} \) is eventually defined on \( D(R) \) and is a critical point of \( E_{\alpha_n} \) with respect to a metric \( \tilde g_n \) on \( D(R) \) converging to the Euclidean metric. The energy \( E(\restr{\tilde s_{\alpha_n}}{D(C_n\delta_n)},\tilde g_n) = E(\restr{\tilde s_{\alpha_n}}{D(y_n,\delta_n)},g_M)\leq B \).

We claim that Proposition prop:2-sacks-uhlenbeck and Theorem thm:5-convergence-crit-map remain correct when \( M=D(R) \) and \( s_\alpha \) are critical maps of \( E_\alpha \) with respect to metrics \( \tilde g_\alpha \) converging to the Euclidean metric. To be precise:

Let \( \tilde s_\alpha:\ (D(R),\tilde g_\alpha) \longrightarrow N\subset \mathbb{R}^k \) be critical maps of \( E_\alpha \) such that

  • \( s_\alpha \) converges weakly to \( s_* \) in \( W^{1,2}(D(R), \text{Euclid}) \),
  • \( E(s_\alpha) < B \)

then there exists at most \( l \) points \( \{x_1,\dots,x_l\} \) in \( \overline{D}(R) \) and a subsequence \( \{s_\beta\} \) such that \( s_\beta \) converges to \( s_* \) in \( C^1(\overline{D}({R/2})\setminus \{x_1,\dots, x_l\}, \mathbb{R}^k) \) uniformly on compacts, and \( s_* \) is harmonic in \( D(R/2) \).

The two ingredients of the proof of Proposition prop:2-sacks-uhlenbeck and Theorem thm:5-convergence-crit-map to be investigated are the covering and the estimate from Lemma lem:main-est. For the estimates, we already remarked that the parameters \( \alpha_0,\epsilon,C(p,D') \) of Lemma lem:main-est can be chosen independent of the metric \( \tilde g_\alpha \) if they are close to Euclidean. For the covering, the investigation is not on the constant \( h \), which can be chosen to be \( 3^{\dim M} \), but on how small the radius of the covering balls must be, but Lemma lem:uni-loc-finite-cover states that their size is dictated by the Ricci curvature and sectional curvature of \( \tilde g_\alpha \), which are also uniformly bounded.

Using Assertion prop:3-star, passing to a subsequence of \( \{\tilde s_{\alpha_n}\} \) if necessary, we can suppose that \( \tilde s_{\alpha_n} \to s_*\) in \( C^1(D(R), \mathbb{R}^k) \). Note that there is no singular point where \( \{\tilde s_{\alpha_n}\} \) fails to converge because \( |d\tilde s_{\alpha_n}| \) is bounded uniformly on \( D(R) \) (hence cannot explode as in Assertion assert:3-star). We can also choose, by a diagonal argument, a subsequence of \( \{\tilde s_{\alpha_n}\} \) that converges to \( s_* \) in \( C^1(\mathbb{R}^2) \) uniformly on compacts.

It is clear that \( s_*: \mathbb{R}^2 \longrightarrow N \) is harmonic and nontrivial because \[ \left| ds_*(0)\right|_{\text{Euclid}} = \lim_{\alpha_{n}\to 1}|d\tilde s_{\alpha_n}(0)|_{\tilde g_{\alpha_n}} = 1. \] Also, \[ \int_{D(R)}|ds_*|^2 dE = \lim_{\alpha_n \to 1}\int_{D(R)} |d\tilde s_{\alpha_n}|^2dV_{\tilde g_\alpha} \leq \limsup_{\alpha\to 1} 2E(\restr{s_{\alpha}}{D(x_1,2\delta_n)}) < 2B \] which means the energy of \( s_* \) on \( \mathbb{R}^2 \) is bounded above by \( 2B \).

Now since \( (\mathbb{R}^2, \text{Euclid}) \) is conformal to \(\mathbb{S}^2\setminus\{p\} \), \( s_* \) can be seen as a harmonic map on \( \mathbb{S}^2\setminus\{p\} \) with the same (finite) energy. By Extension theorem thm:extension-sacks-uhlenbeck, \( s_* \) extends to a nontrivial harmonic map from \( \mathbb{S}^2 \) to \( N \).

  1. We can have a better estimate of \( E(s_*) \). For any \( R>0 \), one has \[ E(\restr{s_*}{D(R)}) + E(\restr{s}{M\setminus \cup_{i=1}^l D(x_i,\delta_n)}) \leq \limsup_{\alpha_n\to 1} \left[E(\restr{s_{\alpha_n}}{D(x_1,\delta_n)}) + E(\restr{s_{\alpha_n}}{M\setminus\cup_{i=1}^l D(x_i,\delta_n)})\right] \] Let \( \delta \to 0 \) then \( R\to +\infty \), one has \[ E(s_*) + E(s) \leq \limsup_{\alpha\to 1} E(s_\alpha). \]
  2. The proof of Theorem thm:6-final-sacks-uhlenbeck also gives a constraint on the image of \( s_* \): since \( s_*(D(R))\subset \overline{\cup_{1<\beta<\alpha}s_\beta(D(x_1,2\delta))} \) for all \( \alpha \) arbitrarily close to \( 1 \) and \( \delta \) arbitrarily small, one has \[ s_*(\mathbb{S}^2)\subset \bigcap_{\delta \to 0}\bigcap_{\alpha \to 1} \overline{\bigcup_{1<\beta<\alpha}s_\beta(D(x_1, \delta))} \]

6 Minimal immersions of \( \mathbb{S}^2 \).

We use the following result:

If \( s: \mathbb{S}^2 \longrightarrow N \) is a nontrivial harmonic map and \( \dim N\geq 3 \), then \( s \) is a \( C^\infty \) conformal, branched, minimal immersion.

The "minimal" part follows from eells_harmonic_1964, the "branched" part follows from gulliver_theory_1973 and the "conformal" part follows from chern_volume_1975 and the fact that there is no nontrivial holomorphic quadratic differential on \( \mathbb{S}^2 \). Theorem thm:6-final-sacks-uhlenbeck gives:

If the universal covering \( \tilde N \) of \( N \) is not contractible then there exists a \( C^\infty \) conformal, branched, minimal immersion \( s: \mathbb{S}^2 \longrightarrow N \).

Bibliography

By Tien NGUYEN MANH.
Last updated 2019-10-22 Tue 18:03.