January 01, 1970
In this paper we classify the low energy \(\varepsilon\)-harmonic maps from the surfaces of constant curvature with positive genus into the round sphere. We find that all such maps with degree \(\pm1\) are all quantitively close to a bubble configuration with bubbles forming at special points on the domain with bubbling radius proportional to \(\varepsilon^{1/4}\).
The Dirichlet energy is one of the most widely studied objects in geometric analysis. Let \((M^2,g)\), \((N^n,h)\) be Riemannian manifolds and \(u\in W^{1,2}(M,N)\). The Dirichlet energy is defined as follows \[E[u]=\int_M|\nabla u|^2.\] Critical points of this energy are known as harmonic maps. The Dirichlet energy functional is not a particularly nice functional, in the sense that regularity theory for it is difficult and it does not satisfy the Palais-Smale compactness conditions, so existence is also difficult.
In order to deal with the existence problem Sacks and Uhlenbeck introduced in [1] the notion of \(\alpha\)-harmonic maps which arise as critical points of a perturbed energy functional. Regularity for these maps is much easier and the functional does indeed satisfy the Palais-Smale conditions, so in particular energy minimisers within homotopy classes exist. As \(\alpha\) tends to 1 their perturbed functional returns to the Dirichlet energy so if we take a sequence of \(\alpha_k\)-harmonic maps \(u_k\) with \(k\rightarrow0\) we hope that this will converge to a fully harmonic map. This does indeed happen weakly in \(W^{1,2}\) and in \(C^\infty_{\text{loc}}\) away from a finite number of points. At these points we observe the standard bubbling phenomenon, when rescaled we obtain harmonic maps from the sphere.
In [2] Lamm introduced the notion of \(\varepsilon\)-energy for \(\varepsilon>0\) and \(u\in W^{2,2}(M,N)\) defined as \[E_{\varepsilon}[u]=\int_M|\nabla u|^2+\varepsilon|\Delta u|^2\] where we further assume that \(N\) is isometrically embedded into some Euclidean space. We call critical points of this energy \(\varepsilon\)-harmonic maps. This again satisfies Palais-Smale and has good regularity theory. Note that the Laplacian here is the extrinsic Laplacian obtained by viewing \(u\) as a map \(M\rightarrow N\hookrightarrow\mathbb{R}^l\) and so the energy depends on the embedding in Euclidean space chosen for \(N\). Using the tension field instead would not give us any benefits as harmonic maps would still be critical points of our ‘intrinsic-\(\varepsilon\)-energy’. This offers the added complications from being a higher order term and the Euler-Lagrange equation becomes a 4th order equation, it however has the added benefit of linearity which offers some benefits in computations.
One of the questions that can be asked about these approximate harmonic maps is which maps can be obtained in the limit. When both the domain and the target the round 2 sphere, \(\mathbb{S}^2\), all harmonic maps, up to reflection, are rational transformations. So in particular they must be Möbius transformations in the degree 1 case. Though in fact it turns out not all of these can be reached by our approximate harmonic maps. Lamm, Malchiodi and Micallef [3],[4] showed that any degree \(\pm1\) \(\alpha\)-harmonic maps with \(\alpha\) sufficiently close to 1 and \(\alpha\)-energy sufficiently small must be a rotation, up to orientation. The same result also holds for \(\varepsilon\)-harmonic maps, in [5] Hörter, Lamm and Micallef showed that any degree \(\pm1\) \(\varepsilon\)-harmonic map with \(\varepsilon\) and \(\varepsilon\)-energy sufficiently small must be a rotation, up to orientation. Gianocca also showed that a similar result holds for the Ginzburg-Landau approximation in [6]. For any \(\gamma>0\) there exists a \(\varepsilon_0\) such that all critical points \(u:\mathbb{S}^2\rightarrow\mathbb{R}^3\) of the Ginzburg-Landau energy with energy \(E^{GL}_\varepsilon[u]<8\pi-\gamma\) must be rotations, up to a scaling and reflection.
We will now study the case where the domain \(\Sigma\) is some Riemannian surface of constant curvature and the target is again round \(\mathbb{S}^2\). The key difference in this case is that there do not exist any degree \(\pm1\) harmonic maps \(u:\Sigma\rightarrow\mathbb{S}^2\). However we can certainly find degree \(\pm1\) \(\varepsilon\)-harmonic maps by taking the energy minimiser within the homotopy class. By taking a sequence of degree 1 \(\varepsilon\)-harmonic maps with \(\varepsilon\) decreasing to 0 and \(\varepsilon\)-energy decreasing to \(4\pi\) we see that we must be in the case of bubble convergence. Due to the fact that our target is the round sphere we can use the energy identity and no neck results proved in [2] and [7]. These allow us to say that we are \(W^{1,2}\) and \(L^\infty\) close to a bubble configuration containing a constant limit map and a harmonic map \(\mathbb{S}^2\rightarrow\mathbb{S}^2\). This gives us a qualitative picture of what \(\varepsilon\)-harmonic maps must look like when \(\varepsilon\) is sufficiently small and \(\varepsilon\)-energy is sufficiently close to \(4\pi\).
This same phenomenon occurs in the \(\alpha\)-harmonic case and in [8] Sharp shows quantitive results about this setup. Indeed he shows that bubbles can only be blown at critical points of a specific function on the domain \(\mathcal{J}\) and that the bubbling radius must be proportional to \(\sqrt{\alpha-1}\), with the exact scale depending on the value of \(\mathcal{J}\) at the blow up point. The proof of this relies on constructing a set of explicit singularity models \(\mathcal{Z}\) which we know we are close to. Then by detailed analysis of the \(\alpha\)-energy on \(\mathcal{Z}\) they obtain their results. This builds on the idea introduced by Malchiodi, Rupflin and Sharp in [9], and further developed by Rupflin in [10] and [11].
In this paper we will show that this approach will also work for \(\varepsilon\)-harmonic maps and establish similar quantitive results in the low energy, degree \(\pm1\) case. To begin we define our domains as well as a special function.
Definition 1. Let \((\Sigma, g)\) be a closed Riemannian surface of genus \(\gamma\geq 1\) equipped with a metric \(g\) of constant curvature zero when \(\gamma=1\) and of constant curvature -1 otherwise. In the flat case we also impose \(\textrm{Area}_g(\Sigma) = 1\). Let \(\{\phi_j\}\) be an arbitrary \(L^2\)-orthonormal basis of holomorphic one-forms on \(\Sigma\) and define \[\mathcal{J} (a) := -2\pi c_\gamma \sum_{j} |\phi_j (a)|^2\] where \(c_1=1\) and when \(\gamma\geq 2\), \(c_\gamma=4\). Note that one can show that \(\mathcal{J}\) is independent of the choice of basis.
This \(\mathcal{J}\) is closely related to the Green’s function on the surface as well as the Bergman kernel, see [9] section 6.
As discussed we know that in our setting we must look like a harmonic bubble blown at a point. This means that we will be able to parametrise our set of approximate bubbles \(\mathcal{Z}\) by the following three variables. \(a\in\Sigma\), the point at which our bubble is blown. \(\lambda\in\mathbb{R}_{>1}\), the scale at which our bubble is blown, \(1/\lambda\) will therefore be the bubbling radius. \(R\in O(3)\), a rotation and choice of orientation of our bubble. We will set \[\pi_\lambda=\Big(\frac{2\lambda x}{1+\lambda^2|x|^2},\frac{1-\lambda^2|x|^2}{1+\lambda^2|x|^2}\Big)\] then we will define \(z=z_{R,\pi,a}\in C^\infty(\Sigma,\mathbb{S}^2)\) to look like \(R\pi_\lambda\) in a neighbourhood of \(a\) in local coordinates and be roughly constant away from \(a\), see section 2 for the detailed construction. This means our singularity solutions will be a 6 dimensional non-compact manifold. In our neighbourhood of \(a\) we have that for large \(\lambda\), \(\pi_\lambda\) roughly looks like \[\Big(\frac{2x}{\lambda|x|^2},1\Big)\approx\Big(\frac{2}{\lambda}\frac{\partial}{\partial x}\log|x|,1\Big).\] The Greens function near \(a\) in local coordinates looks like \(\log|x|\) + some error term so smoothing out to the roughly constant section using the Greens function is a natural choice.
To motivate our results we will expand the \(\varepsilon\)-energy along \(\mathcal{Z}\). We will have, similarly to the expansions in [9] and [8], for \(z\in\mathcal{Z}\) \[E_\varepsilon[z]=4\pi-4\pi\mathcal{J}(a)\frac{1}{\lambda^2}+\frac{32\pi}{3c_\gamma}\varepsilon\lambda^2+\mathcal{O}(\frac{1}{\lambda^3})+\mathcal{O}(\varepsilon).\] We shall choose \(z\) to close to our critical point of \(E_\varepsilon\), so differentiating with respect to \(\lambda\) and \(a\) should give us something small. In particular one can show that \[\partial_\lambda E_\varepsilon[z]=8\pi\mathcal{J}(a)\frac{1}{\lambda^3}+\frac{64\pi}{3c_\gamma}\varepsilon\lambda+\mathcal{O}(\frac{1}{\lambda^4})+\mathcal{O}(\frac{\varepsilon}{\lambda})\] and, in the hyperbolic case, \[\nabla_A E_\varepsilon[z]=4\pi\nabla_A\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3})+\mathcal{O}(\varepsilon\lambda).\] Where \(\nabla_A\) is the derivative of the base point \(a\) in the direction \(A\). These align with naïvely differentiating the energy. We will show that these quantities are indeed quantitively small which gives us information about the bubble scale and location.
The final element we need to introduce is the ‘\(z\)-norm’ (\(\|\cdot\|_z\)) first introduced by Rupflin [10]. We already know that our maps are converging to a constant in an \(L^2\) sense so we need to introduce a factor which blows up near the bubbles so we can see what happens near them. Further, Rupflin showed that the Dirichlet energy is non-degenerate with respect to this norm orthogonally to \(\mathcal{Z}\) under certain conditions which will be very important. A difficulty occurs when using this norm in our setting as it does not involve any second order terms. We are able to overcome this by leveraging the smallness of \(\varepsilon\).
We now state the main two theorems of the paper.
Theorem 2. Let \(\Sigma\) be as in 1. There exist \(\varepsilon_0(\Sigma),\delta_0(\Sigma),C(\Sigma)>0\) such that if \(0<\varepsilon<\varepsilon_0\), \(u:\Sigma\rightarrow\mathbb{S}^2\) is a \(\varepsilon\)-harmonic map with \(E_\varepsilon[u]\leq4\pi+\delta_0\) then either
\(u\) has degree zero and \(\|\nabla u\|_{L^\infty(\Sigma)}\leq C\), or
\(u\) has degree \(\pm 1\) and there exists \(z(u)\) realising inf\(\{\|u-z\|_z: z\in\mathcal{Z}\}\), for any such \(z(u)\), setting \(a=a(z(u))\) and \(\lambda=\lambda(z(u))\), we have \[|\mathcal{J}(a)\frac{1}{\lambda^4}+\frac{8}{3c_\gamma}\varepsilon|\leq C\varepsilon^{3/2}|\log\varepsilon|.\]
Theorem 3. Let \(\Sigma\) be as in 1 with genus \(\gamma\geq2\). There exist \(\varepsilon_0(\Sigma),\delta_0(\Sigma),C(\Sigma)>0\) such that if \(0<\varepsilon<\varepsilon_0\), \(u:\Sigma\rightarrow\mathbb{S}^2\) is a \(\varepsilon\)-harmonic map with degree \(\pm1\) and \(E_\varepsilon[u]\leq4\pi+\delta_0\) then for any \(z(u)\) realising inf\(\{\|u-z\|_z: z\in\mathcal{Z}\}\) with \(a=a(z(u))\) we have \[\nabla_A\mathcal{J}(a)\leq C\varepsilon^{1/4}|\log\varepsilon|.\]
These theorems immediately give us an explicit picture of what must happen in the limit.
Corollary 1. Let \(u_k:\Sigma\rightarrow\mathbb{S}^2\) be a sequence of \(\varepsilon_k\)-harmonic maps with \(0<\varepsilon_k\downarrow0\) and \(E_{\varepsilon_k}[u_k]\rightarrow\Lambda\leq4\pi\) then there exists a subsequence such that either
\(u_k\) all have degree zero and converge smoothly to some degree zero harmonic map \(u:\Sigma\rightarrow\mathbb{S}^2\), or
\(u_k\) all have degree 1 or all have degree -1 and converge in the standard bubbling sense to a bubble tree with the limit map constant and one bubble which is blown at a critical point \(a_c\) of \(\mathcal{J}\) at a bubbling rate of \[r_k=\Big(\frac{8\varepsilon_k}{3c_\gamma}|\mathcal{J}(a)|^{-1}\Big)^{1/4}.\]
Acknowledgements
The author would like to thank his supervisor Ben Sharp for many helpful discussions. The author is funded by Engineering and Physical Sciences Research Council (EPSRC) - EP/W524372/1, Studentship 2927009.
We start by explicitly defining \(\mathcal{Z}\), our space of singularity models as in [10] and [8]. With \((\Sigma,g)\) as in Definition 1 set \(\iota=\frac{1}{2}\text{inj}(\Sigma,g)\), half of the injectivity radius. In the hyperbolic case set \(\rho=\tanh(\iota)\) and note that for any \(a\in\Sigma\) there exists an orientation preserving isometric isomorphism \[F_a:(B_{2\iota}(a),g)\rightarrow(\mathbb{D}_\rho,\frac{4}{(1-|x|^2)^2}g_E)\] with \(F_a(a)=0\), \(\mathbb{D}_\rho=\{x\in\mathbb{R}^2:|x|<\rho\}\) and \(g_E\) is the Euclidean metric. In the flat case we simply take \(\rho=2\iota\) and \(F_a\) to be some choice of flat orientation preserving chart. We will fix the Greens function on \(\Sigma\) which solves \[-\Delta_pG(p,q)=2\pi\delta_q-\frac{2\pi}{\text{Vol}(\Sigma)}.\] From now on work with local coordinates, for \(p,q\in B_{2\iota}(a)\) we write \(x=F_a(p)\) and \(y=F_a(q)\). In these coordinates we can locally write the Greens function as \[G(p,q)=G_a(x,y)=-\log|x-y|+J_a(x,y),\] where \(J_a\) is some smooth function, dependent on the choice of \(F_a\). Set \(r=\rho/4\), then fix some \(\phi\in C_c^\infty(\mathbb{D}_{2r},[0,1])\) radial with \(\phi\equiv 1\) on \(\mathbb{D}_r\). Now define for \(\lambda\geq1\) \[\tilde{z}_{\lambda,a}(p)=\begin{cases} \hat{z}_{\lambda,a}(F_a(p))&\text{if }p\in B_\iota(a)\\ (\frac{2}{\lambda}(\partial_{q^1}G(p,a)-\partial_{y^1}J_a(0,0)),\frac{2}{\lambda}(\partial_{q^2}G(p,a)-\partial_{y^2}J_a(0,0)),-1)&\text{if }p\notin B_\iota(a) \end{cases}\] where, in our local coordinates \[\begin{align} \hat{z}_{\lambda,a}(x)=&\phi(x)\big(\pi_\lambda(x)+(\frac{2}{\lambda}(\nabla_yJ_a(x,0)-\nabla_yJ_a(0,0)),0)\big)\\ &+(1-\phi(x))(\frac{2}{\lambda}(\nabla_yG_a(x,0)-\nabla_yJ_a(0,0)),-1) \end{align}\] with \(\pi_\lambda(x)=\big(\frac{2\lambda x}{1+\lambda^2|x|^2},\frac{1-\lambda^2|x|^2}{1+\lambda^2|x|^2}\big)\), the stereographic projection at scale \(\lambda\). Finally set \[\begin{align} z_{\lambda,a}=&P_{\mathbb{S}^2}(\tilde{z}_{\lambda,a})=\frac{\tilde{z}_{\lambda,a}}{|\tilde{z}_{\lambda,a}|}\\ \mathcal{Z}=&\{Rz_{\lambda,a}|a\in\Sigma,R\in O(3),\lambda>1\}. \end{align}\] Also, in our local coordinates, define \[j(x)=j_{\lambda,a}(x):=(\frac{2}{\lambda}(\nabla_yJ_a(x,0)-\nabla_yJ_a(0,0)),0).\] So in \(\mathbb{D}_{r}\) we have \(z=P_{\mathbb{S}^2}(\pi+j)\). We will also frequently use the notation for \(t\leq 2r\) \[U_t:=F_a^{-1}(\mathbb{D}_t).\]
This gives us our family of singularity models which look like a single bubble forming at a point. We will now formally define the norm discussed in the introduction as introduced by Rupflin in [10].
Definition 4. Given \(V,W\in W^{1,2}(\Sigma,\mathbb{R}^3)\) and \(z\in\mathcal{Z}\) define an inner product \[\langle V,W\rangle_z=\int_\Sigma\nabla V\cdot\nabla W+\rho^2_zV\cdot W\textrm{d}\Sigma\] where \[\rho_z(p):=\begin{cases} \frac{\lambda}{1+\lambda^2\textrm{d}_g(p,a)^2} &\text{if }p\in B_\iota(a)\\ \frac{\lambda}{1+\lambda^2\iota^2} &\text{if }p\in\Sigma\setminus B_\iota(a). \end{cases}\] As usual we write \(\|V\|_z:=\langle V,V\rangle_z^{1/2}\).
The factor \(\rho\) rescales the \(L^2\) norm around the bubbling point to the bubble scale, which allows detailed analysis near it.
We also mention a bound shown in [10], which allows us to turn on \(L^2\) bounds or an integral around a boundary into a ‘\(z\)-norm’ bound at some cost in \(\lambda\).
Remark 5. [[10] (2.28) and Appendix B] For any \(s\in[1,\infty)\) there exists \(K(s,\Sigma)\) such that \[|\int_{\partial B_\iota(a)}v\textrm{d}\Sigma|+\|v\|_{L^s(\Sigma)}\leq K(\log\lambda)^{1/2}\|v\|_z.\]
To begin with we expand the \(\varepsilon\)-energy along \(\mathcal{Z}\). For this and many other of our results we will require very detailed estimates on \(z\) and its derivatives. We have deferred these to Appendix 7.
Lemma 1. For \(z\in \mathcal{Z}\) and \(0<\varepsilon<1\) we have \[E_\varepsilon[z]=4\pi-4\pi\mathcal{J}(a)\frac{1}{\lambda^2}+\frac{32\pi}{3c_\gamma}(\varepsilon\lambda^2)+\mathcal{O}(\frac{1}{\lambda^3})+\mathcal{O}(\varepsilon).\]
Proof. For the Dirichlet part see [9] or appendix 8. \[E[z]=4\pi-4\pi\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3}).\] For the biharmonic part we have, using 27 , \[\begin{align} \varepsilon\int_{\Sigma}|\Delta z|^2=&\varepsilon\int_{\mathbb{D}_{r}}|\Delta^g z|^2\textrm{d}x_g+\mathcal{O}(\varepsilon). \end{align}\] Then from 26 \[|\Delta^g z|^2=\frac{1}{c_\gamma^2}|\Delta \pi_\lambda|^2+\mathcal{O}(\frac{\lambda^2}{(1+\lambda^2|x|^2)^2}),\] then using 22 , \[\begin{align} \begin{aligned} \int_{\mathbb{D}_r}\frac{1}{c_\gamma^2}|\Delta \pi_\lambda|^2\text{d}x_g=& \int_{\mathbb{D}_r}\frac{64\lambda^4}{(1+\lambda^2|x|^2)^4}(\frac{1}{c_\gamma}+\mathcal{O}(|x|^2))dx= \frac{64\lambda^2\pi}{3c_\gamma}+\mathcal{O}(1)\\ \int_{\mathbb{D}_r}\mathcal{O}(\frac{\lambda^2}{(1+\lambda^2|x|^2)^2})\text{d}x_g=&\mathcal{O}(1). \end{aligned} \end{align}\] Combining these integrals completes the proof. ◻
We also note here that, for \(u\in W^{2,2}(\Sigma,\mathbb{S}^2)\) and \(V,W\in W^{2,2}(\Sigma,\mathbb{R}^3)\) with \(V(u(x)),W(u(x))\in T_{u(x)}\mathbb{S}^2\) almost everywhere, we have \[\begin{align}\label{eq:32variation32expansions} \text{d}E_\varepsilon(u)[V]=&\int_\Sigma\nabla u\cdot\nabla V+\varepsilon\int_\Sigma\Delta u\cdot\Delta V\\ \text{d}^2E_\varepsilon(u)[V,W]=&\int_\Sigma\nabla V\cdot\nabla W-|\nabla u|^2(V\cdot W) +\varepsilon\int_\Sigma\Delta V\cdot\Delta W-\Delta u\cdot\Delta(u(V\cdot W)) \end{align}\tag{1}\]
We mention the energy identity and no-neck results for \(\varepsilon\)-harmonic maps into the round 2-sphere, noting again that the sphere must be embedded equivariantly into \(\mathbb{R}^3\). These allow us to say that our \(\varepsilon\)-harmonic maps do look like our models in an \(W^{1,2}\) sense and in an \(L^\infty\) sense.
Theorem 6. [[2] Theorem 1.1, [7] Theorem 1.5] Let \(\Sigma\) be as in Definition 1 and \(\mathbb{S}^2\hookrightarrow\mathbb{R}^3\) be the standard embedding. If \((u_k)\in W^{2,2}(\Sigma,\mathbb{S}^2)\) is a sequence of \(\varepsilon_k\) harmonic maps with \(0<\varepsilon_k\downarrow0\) and \(E_{\varepsilon_k}[u_k]\) uniformly bounded, then along some subsequence there exist a smooth harmonic map \(u_\infty:\Sigma\rightarrow\mathbb{S}^2,\) a finite non negative number of smooth harmonic maps \((\omega^j):\mathbb{R}^2\rightarrow\mathbb{S}^2\) and corresponding sequences \((a^j_k,r^j_k)\) with \(a^j_k\rightarrow a^j\in \Sigma\) and \(r^j_k\rightarrow 0\) such that
\(E_{\varepsilon_k}[u_k]\rightarrow E[u_\infty]+\sum\limits_{k}E[\omega_k],\)
\(\|u-u_\infty-\phi^j(\cdot)\sum\limits_{k}(\omega^j((r^j_k)^{-1}F_{a^j_k}(\cdot))-\omega^j(\infty))\|_{L^\infty(B_\iota(a^j))}\quad\) for each \(j\)
where \(\phi^j\in C^\infty(\Sigma)\) is defined by \(\phi^j(p)=\phi(F_{a^j}(p))\) for some fixed \(\phi\in C_c(\mathbb{D}_t,[0,1])\) with \(\phi\equiv 1\) on \(\mathbb{D}_{t/2}\), for \(t\) fixed with \(t<\rho/2,\textrm{min}_{j_1\neq j_2}\{|a^{j_1}-a^{j_2}|/2\}\).
In [2] Lamm showed a low energy regularity theorem for \(\varepsilon\) harmonic maps which we recall. Due to our low energy setting this will entail pointwise estimates on the neck region.
Lemma 2. [[2] Corollary 2.10] There exist \(\delta_0>0\) and \(C>0\) such that if \(u_\varepsilon\in C^\infty(\Sigma,\mathbb{S}^2)\) is \(\varepsilon\)-harmonic with \(E_\varepsilon(u_\varepsilon,B_{32R}(x_0))<\delta_0\) for some \(x\in\Sigma\) and \(R>0\). Then for all \(\varepsilon>0\) sufficiently small and any \(k\in \mathbb{N}\) \[\sum_{i=1}^kR^i\|\nabla^iu_\varepsilon\|_{L^\infty(B_R(x_0))}\leq C\sqrt{E_\varepsilon(u_\varepsilon,B_{32R}(x_0))}.\]
We now start by clarifying the degree 0 case.
Lemma 3. There exists \(\varepsilon_0(\Sigma)>0,\delta_0(\Sigma)>0,C(\Sigma)>0\) such that if \(0<\varepsilon<\varepsilon_0\), \(u:\Sigma\rightarrow \mathbb{S}^2\) is a degree zero \(\varepsilon\)-harmonic map with \(E_\varepsilon[u]\leq 4\pi+\delta_0\) then \(\|\nabla u\|_{L^\infty(\Sigma)}\leq C\)
Proof. Suppose not, then there exists a sequence \((u_k)\) of \(\varepsilon_k\)-harmonic maps with degree zero with \(\varepsilon_k\downarrow0\) and \(E_{\varepsilon_k}[u_k]\rightarrow 4\pi\), but \(\|\nabla u\|_{L^\infty(\Sigma)}\rightarrow\infty\). This means that we cannot have smooth convergence of \((u_k)\), so along a subsequence we must have bubble convergence to a limit with at least one bubble. By consideration of energy we must have a bubble of degree \(\pm1\) and the limit map \(u_\infty\) constant. But due to the no neck property in Theorem 6 this convergence must preserve homotopy which contradicts \(u_k\) being degree 0. ◻
In the degree 1 case, as discussed, any \(\varepsilon\)-harmonic map should be \(W^{1,2}\) and \(L^\infty\) close to something in \(\mathcal{Z}\). We prove this and establish a quantitive result.
Lemma 4. For any \(\gamma>0\) there exist \(\varepsilon_0(\Sigma),\delta_0(\Sigma)>0\) such that if \(0<\varepsilon<\varepsilon_0\) and \(u:\Sigma\rightarrow\mathbb{S}^2\) a degree one \(\varepsilon\) harmonic map with \(E_\varepsilon[u]\leq4\pi+\delta_0\) then we have the following
\(\|\nabla u\|_{L^\infty}>\frac{1}{\gamma},\)
\(\exists z\in \mathcal{Z}\) such that \(\|u-z\|_{W^{1,2}}+\|u-z\|_{L^\infty}\leq \gamma,\)
There exists \(z\in\mathcal{Z}\) with \(\|u-z\|_z=\inf_{\eta\in\mathcal{Z}}\{\|u-\eta\|_\eta\}\). For this \(z\) we then have \(\|u-z\|_{z}+\|u-z\|_{L^\infty}\leq \gamma\).
Proof. Take any collection of \(u_k\), degree 1 \(\varepsilon_k\)-harmonic maps with \(\varepsilon_k\downarrow0\) and \(E_{\varepsilon_k}[u_k]\downarrow 4\pi\). If \(\|\nabla u_k\|_{L^\infty}+\varepsilon_k\|\Delta u_k\|_{L^\infty}\leq\frac{1}{\gamma}\) for infinitely many \(k\) then Lemma 2 implies that along a subsequence we must have smooth convergence to a smooth degree one harmonic map \(u_\infty:\Sigma\rightarrow\mathbb{S}^2\) with energy \(4\pi\), which we know does not exist. This means we must be in the case of bubble convergence to precisely one bubble. So we must have \(\|\nabla u_k\|_{L^\infty}+\varepsilon_k\|\Delta u_k\|_{L^\infty}>\frac{1}{\gamma}\) for any \(k\) large enough. Results from [2] now imply that \(\varepsilon_k\|\Delta u_k\|_{L^\infty}\leq C\frac{\varepsilon_k}{r_k^2}\rightarrow 0\), where \(r_k\) is the bubble scale of \(u_k\). It is clear that we must be able to find \(\varepsilon_0,\delta_0\) such that 1. follows.
By the energy identity (Theorem 6), we must have that the limit map, \(u_\infty\), is constant and that the bubble, \(\omega\), is a harmonic map \(\mathbb{S}^2\rightarrow\mathbb{S}^2\) with energy \(4\pi\). This means that, after possibly rotating the domain and target, \(\omega\) must be of the form \(\pi_{\mu}\) for some \(\mu\) fixed, meaning the stereographic projection at scale \(\mu\). Now consider \(z_k:=z_{\mu r_k^{-1},a_k}\) where \(r_k\) is the bubbling radius and \(a_k\) the bubbling point of \(u_k\). Now \(z_k\) must also converge in a \(W^{1,2}\) and \(L^\infty\) sense to the same bubble configuration as \(u_k\). So using both the energy identity and the no neck property we must have \[E[u_k-z_k]+||u_k-z_k||_{L^\infty}\rightarrow0.\] and so 2. must hold.
Taking the \(z\) from 2. the bubble scale converges to 0 so \(\mu r_k^{-1}\rightarrow\infty\), this means that for \(k\) large we can apply Lemma 4.2 from [10]. For \(k\) large enough, there exist \(\zeta_{k}\in\mathcal{Z}\) minimising \(\| u_k-\eta\|_\eta\) over \(\eta\in\mathcal{Z}\) with \(\|u_k-\zeta_{k}\|_{L^\infty}\rightarrow0\). We then have \(\| u_k-\zeta_k\|_{\zeta_k}\leq C\| u_k-z_k\|_{z_k}\rightarrow0\) completing the proof. ◻
Now we have shown that \(u\) must be close to some \(z\) we can start to pass some of the properties from \(z\) onto \(u\). The following lemma says that we can in some sense quantitively think of \(\lambda\) as the inverse of the bubbling radius. These bounds would follow easily from Lemma 2 if \(\lambda\) were replaced by the inverse of some choice of the bubbling radius.
Lemma 5.
There exists \(\varepsilon_0(\Sigma),\delta_0(\Sigma)\) such that if \(0<\varepsilon<\varepsilon_0\), \(u:\Sigma\rightarrow \mathbb{S}^2\) is a degree 1 \(\varepsilon\)-harmonic map with \(E_\varepsilon[u]-4\pi\leq \delta_0\) and \(z\in \mathcal{Z}\) with \(\|u-z\|_z^2\leq \delta_0\) then
\(\|\nabla^k u\|_{L^\infty(\Sigma)}\leq C(k,\Sigma)\lambda^k\) for \(k\geq 0,\)
\(\|\nabla^k u\|_{L^2(\Sigma)}\leq C(k,\Sigma)\lambda^{k-1}\) for \(k\geq 1,\)
\(\varepsilon\lambda^2\leq C(\Sigma)(E_\varepsilon[u]-4\pi)\leq C(\Sigma)\delta_0.\)
Proof. In \(\Sigma\setminus U_{r/4}\), where \(r\) is as in the definition of \(z\), we have \(|\nabla z|\leq C\), with \(C\) independent of \(z\). We can then choose some \(\gamma>0\) small enough such that \(\|\nabla z\|_{L^2(B_\gamma(p))}^2\leq \delta_0\) for any \(p\in\Sigma\setminus U_{r/2}\). We then have \[E_\varepsilon[u;B_\gamma(p)]\leq 2\|\nabla z\|_{L^2(B_\gamma(p))}^2+2\|u-z\|_z^2 +\varepsilon\|\Delta u\|_{L^2(\Sigma)}^2\leq5 \delta_0\] Now we can take \(\delta_0,\varepsilon_0\) small enough such that we may apply Lemma 2. This means that \(\|\nabla^k u\|_{L^\infty(B_{\gamma/32}(p))}\leq C\gamma^{-k}\leq C(k,\Sigma)\), giving our \(L^\infty\) bound on \(\Sigma\setminus U_{r/2}\). In \(U_r\) we have \(|\nabla z|\leq C\frac{\lambda}{1+\lambda^2|x|^2}\) using 25 . By setting \(\hat{z}(x)=z(x/\lambda)\), defined in our local coordinates on \(\mathbb{D}_{r\lambda}(0)\), we again have \(|\nabla \hat{z}|\leq C(\Sigma)\). So we may again choose some \(\gamma>0\) small enough such that \(\|\nabla \hat{z}\|_{L^2(\mathbb{D}_{\gamma}(y))}^2\leq \delta_0\) for any \(y\in \mathbb{D}_{\lambda r/2}(0)\). So for any \(x\in \mathbb{D}_{r/2}\) we have \[E_\varepsilon[u;\mathbb{D}_{\lambda^{-1}\gamma}(x)]\leq 2\|\nabla z\|_{L^2(\mathbb{D}_{\lambda^{-1}r}(x))}^2+2\|u-z\|_z^2 +\varepsilon\|\Delta u\|_{L^2(\Sigma)}^2\leq 5\delta_0\] by scale invariance of energy. Then by Lemma 2 we have \(\|\nabla^k u\|_{L^\infty(\mathbb{D}_{\lambda^{-1}\gamma /32}(x))}\leq C\gamma^{-k}\lambda^k\leq C(k,\Sigma)\lambda^k\), completing our \(L^\infty\) bound.
For the \(L^2\) bound now consider the annulus \(A=\mathbb{D}_r\setminus \mathbb{D}_{T\lambda^{-1}}\) for some \(T>0\) to be decided. Using 25 we get \[\|\nabla z\|_{L^2(A)}^2\leq C\int_{B_r(0)\setminus B_{T\lambda^{-1}}(0)}\frac{\lambda^2}{(1+\lambda^2|x|^2)^2}\leq C\int_{\mathbb{R}^2\setminus B_{T}(0)}\frac{1}{(1+|x|^2)^2}\leq \delta_0\] by fixing \(T(\Sigma)\) large enough. As before this implies that \(E_\varepsilon[u;A]\leq 5 \delta_0\). Now for any \(x\in \mathbb{D}_{r/2}\setminus \mathbb{D}_{2T\lambda^{-1}}\) we have that \(\mathbb{D}_{|x|/2}(x)\subset A\) and so by Lemma 2 we have \(\|\nabla^k u\|_{L^\infty(B_{|x|/64}({x}))}\leq C(k,\Sigma)|x|^{-k}\). This then gives \[\begin{align} \|\nabla^k u\|^2_{L^2(\Sigma)}&\leq C\bigg(\int_{\Sigma\setminus U_{R/2}(a)} + \int_{B_{R/2}(a)\setminus B_{2T\lambda^{-1}}(a)}\frac{1}{|x|^{2k}}+\int_{ B_{2T\lambda^{-1}}(a)}\lambda^{2k}\bigg)\\ &\leq C\lambda^{2k-2} \end{align}\] completing the \(L^2\) bound. The \(L^2\) bound then immediately implies that \[\label{eq:32Delta32into32z32norm} \|\Delta (u-z)\|_{L^2}^2=\int_\Sigma\nabla(u-z)\cdot\nabla\Delta(u-z)\leq\|u-z\|_z\|\nabla\Delta(u-z)\|_{L^2}\leq C\lambda^2\|u-z\|_z\tag{2}\] which allows us to write, using the computations from Lemma 1, \[E_\varepsilon[u]-4\pi\geq \varepsilon\|\Delta u\|_{L^2}^2\geq \varepsilon(\frac{1}{2}\|\Delta z\|_{L^2}^2-\|\Delta (u-z)\|_{L^2}^2)\geq \varepsilon(C_1\lambda^2-C_2\lambda^2\delta_0^{1/2})\] which completes the final bound by taking \(\delta_0\) small enough. ◻
2 will be essential later on as it allows us to bound 2nd order terms in terms of the \(z\)-norm.
Lemma 6. There exist \(\varepsilon_0(\Sigma),\delta_0(\Sigma),c_0>0\) such that if \(0<\varepsilon<\varepsilon_0\), \(u:\Sigma\rightarrow\mathbb{S}^2\) is a degree one \(\varepsilon\)-harmonic map with \(E_\varepsilon\leq4\pi+\delta_0\) and \(z\in\mathcal{Z}\) realises \(\|u-z\|_z=\inf_{\eta\in\mathcal{Z}}\{\|u-\eta\|_\eta\}\) them setting \(w=u-z\) and \(W=\text{d}P(z)[w]\) gives \[\textrm{d}^2E_\varepsilon(z)[W,W]\geq c_0\|w\|_z^2\]
Proof. We have from 1 \[\textrm{d}^2E_\varepsilon(z)[W,W]=\textrm{d}^2E(z)[W,W]+\varepsilon\int|\Delta W|^2-(\Delta^2z\cdot z)|W|^2\] From Lemma 3.2 in [10] we have that, when \(\lambda(z)\) is large enough, all eigenvalues of \(\text{d}^2E\) are bounded away from 0 on \(T_z\mathcal{Z}^{\perp_z}\), the orthogonal complement of \(T_z\mathcal{Z}\) with respect to the \(\langle\cdot,\cdot\rangle_z\) inner product. We can guarantee \(\lambda\) is large enough by Lemma 4 and Lemma 5. We also know that these eigenvalues must be positive as the limiting bubble is stable. Now from Lemma 4.3 in [10] we have that \(\|W^{\top_{T_z\mathcal{Z}}}\|\leq C\|w\|_{L^\infty}\|w\|_z\) which gives, using Lemma 4 to ensure \(\|w\|_{L^\infty}\) small enough, \[\|W^{\bot_{T_z\mathcal{Z}}}\|^2_z\geq \|W\|^2_z-C\|w\|^2_{L^\infty}\|w\|^2_z\geq \frac{1}{4}\|w\|^2_z\] also using [eq:32w32and32W32equiv46]. We also note that \[d^2E(z)[A,B]=\int_\Sigma\nabla A\cdot\nabla B-|\nabla z|^2(A\cdot B)\leq C\|A\|_z\|B\|_z\] Let \(c_0\) be the lowest eigenvalue of \(E(z)\) on \(T_z\mathcal{Z}^{\perp_z}\). We can then calculate, using Young’s inequality on the second term, \[\begin{align} d^2E(z)[W,W]=&d^2E(z)[W^{\bot_{T_z\mathcal{Z}}},W^{\bot_{T_z\mathcal{Z}}}]+2d^2E(z)[W^{\bot_{T_z\mathcal{Z}}},W^{\top_{T_z\mathcal{Z}}}]+d^2E(z)[W^{\top_{T_z\mathcal{Z}}},W^{\top_{T_z\mathcal{Z}}}]\\ \geq& \frac{c_0}{2}\|W^{\bot_{T_z\mathcal{Z}}}\|^2_z- C\|W^{\top_{T_z\mathcal{Z}}}\|^2_z\\ \geq& \frac{c_0}{16}\|w\|^2_z \end{align}\] by taking \(\varepsilon_0,\delta_0\) small enough. Using 33 we have \(|\Delta^2z\cdot z|\leq C\lambda^2\rho_z^2\) which gives, further using [eq:32w32and32W32equiv46] and Lemma 5, \[\varepsilon\int|\Delta W|^2-(\Delta^2z\cdot z)|W|^2\geq -C\varepsilon\lambda^2\|W\|_z^2\geq-\frac{c_0}{32}\|w\|^2_z\] by taking \(\varepsilon_0,\delta_0\) small enough. This completes the result. ◻
We start by presenting proofs of the main two theorems, these rely on many extra bounds from the final two sections.
Proof of Theorem 2. Take \(\varepsilon_0,\delta_0>0\) small, to be determined. Now take \(u\) some degree \(1\) \(\varepsilon\)-harmonic map with \(0<\varepsilon<\varepsilon_0\) and \(E_\varepsilon[u]\leq 4\pi+\delta_0\). Then by Lemma 4 there exists some \(z\in\mathcal{Z}\) with \(\|u-z\|_z=\inf_{\eta\in\mathcal{Z}}\{\|u-\eta\|_\eta\}\) by taking \(\varepsilon_0,\delta_0\) small enough. Set \(w=u-z\), \(u_t=P(z+t(u-z))\) and \(W_t=\partial_tu_t\), in particular set \(W:=W_0=\textrm{d}P(z)[w]\). This gives \[\begin{align} 0=\textrm{d}E_\varepsilon(u)[W_1]=&\int_0^1\frac{\partial}{\partial t}\textrm{d}E_\varepsilon(u_t)[W_t]+\textrm{d}E_\varepsilon(z)[W]\\ =&\int_0^1\textrm{d}^2E_\varepsilon(u_t)[W_t,W_t]-\textrm{d}^2E_\varepsilon(z)[W,W]+\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)\partial_tW_t] \\&+\textrm{d}^2E_\varepsilon(z)[W,W]+\textrm{d}E_\varepsilon(z)[W].\\ \end{align}\] Using Lemma 6 we have \(\textrm{d}^2E_\varepsilon(z)[W,W]\geq c_0\|w\|_z^2\). Using Lemma 11 as well as Lemma 4, Lemma 5 and 2 we have \[\begin{align} |\textrm{d}^2E_\varepsilon(u_t)[W_t,W_t]-\textrm{d}^2E_\varepsilon(z)[W,W]|+|\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)\partial_tW_t]|\\ \leq C((\|w\|_{L^\infty}+\varepsilon\lambda^2+\varepsilon\|\nabla w\|_{L^{\infty}}^2)\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2)\\ \leq \frac{c_0}{2}\|w\|^2_z+C\varepsilon\lambda^2\|w\|_z \end{align}\] for \(\varepsilon_0,\delta_0\) small enough. We also have using Lemma 10 and [eq:32w32and32W32equiv46] \[\textrm{d}E_\varepsilon(z)[W]\leq C(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|w\|_z.\] Combining these gives \[\label{eq:32w32z32norm32bound} \|w\|_z\leq C(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2).\tag{3}\] For a general \(T\in T_z\mathcal{Z}\) with \(T_t=\textrm{d}P(u_t)[T]\) \[\begin{align} 0=\textrm{d}E_\varepsilon(u)[T_1]=&\int_0^1\frac{\partial}{\partial t}\textrm{d}E_\varepsilon(u_t)[T_t]+\textrm{d}E_\varepsilon(z)[T]\\ =&\int_0^1\textrm{d}^2E_\varepsilon(u_t)[T_t,W_t]-\textrm{d}^2E_\varepsilon(z)[T,W]+\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)\partial_tT_t] \\&+\textrm{d}^2E_\varepsilon(z)[T,W]+\textrm{d}E_\varepsilon(z)[T].\\ \end{align}\] Setting \(T=\partial_\lambda z\) then using 2 , 3 , Lemma 10, Lemma 12 and Lemma 5 gives, for \(\varepsilon_0,\delta_0\) small enough, \[\begin{align} |\textrm{d}^2E_\varepsilon(u_t)[T_t,W_t]-\textrm{d}^2E_\varepsilon(z)[T,W]|+|\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)\partial_tT_t]|+|\textrm{d}^2E_\varepsilon(z)[T,W]|\\ \leq C\frac{1}{\lambda}(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^2})\|w\|_z^2+\varepsilon\|\Delta w\|^2_{L^2})+C\frac{1}{\lambda}(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|w\|_z\\ \leq C(\frac{\log\lambda}{\lambda^5}+\varepsilon\frac{(\log\lambda)^{1/2}}{\lambda}+\varepsilon^2\lambda^3+\varepsilon^3\lambda^5). \end{align}\] From Lemma 7 we have \[\textrm{d}E_\varepsilon(z)[T]=\partial_\lambda E_\varepsilon[z]=8\pi\mathcal{J}(a)\frac{1}{\lambda^3}+\frac{64\pi}{3c_\gamma}\varepsilon\lambda+\mathcal{O}(\frac{1}{\lambda^4})+\mathcal{O}(\frac{\varepsilon}{\lambda}).\] So combining we get \[8\pi\mathcal{J}(a)\frac{1}{\lambda^3}+\frac{64\pi}{3c_\gamma}\varepsilon\lambda=\mathcal{O}(\frac{\log\lambda}{\lambda^5})+\mathcal{O}(\varepsilon\frac{(\log\lambda)^{1/2}}{\lambda})+\mathcal{O}(\varepsilon^2\lambda^3)+\mathcal{O}(\varepsilon^3\lambda^5).\] We may assume \(\varepsilon\lambda^2\) is as small as we like from Lemma 5 which then gives the existence of some \(C\) such that \[\label{eq:32epsilon32to32lambda32ratio} C^{-1}\frac{1}{\lambda^4}\leq\varepsilon\leq C\frac{1}{\lambda^4}\tag{4}\] and so we obtain \[|\mathcal{J}(a)\frac{1}{\lambda^4}+\frac{8}{3c_\gamma}\varepsilon|=\mathcal{O}(\frac{\log\lambda}{\lambda^6})=\mathcal{O}(\varepsilon^{3/2}|\log\varepsilon|).\] ◻
Proof of Theorem 3. Under these same conditions now set \(T=\nabla_A z\). Using 4 as well as 2 , Lemma 10, Lemma 12 and Lemma 5 we get \[\begin{align} |\textrm{d}^2E_\varepsilon(u_t)[T_t,W_t]-\textrm{d}^2E_\varepsilon(z)[T,W]|+|\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)\partial_tT_t]|+|\textrm{d}^2E_\varepsilon(z)[T,W]|\\ \leq C\lambda(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^2})\|w\|_z^2+\varepsilon\|\Delta w\|^2_{L^2})+C\lambda(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|w\|_z\\ \leq C\frac{\log\lambda}{\lambda^3} \end{align}\] for \(\varepsilon_0,\delta_0\) small enough. Lemma 8 in combination with 4 says that \[\begin{align} \nabla_A E_\varepsilon[z]=&4\pi\nabla_A\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3})+\mathcal{O}(\varepsilon\lambda)\\ =&4\pi\nabla_A\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3}). \end{align}\] So combining these equations we get \[\nabla_A\mathcal{J}(a)=\mathcal{O}(\frac{\log\lambda}{\lambda})=\mathcal{O}(\varepsilon^{1/4}|\log\varepsilon|).\] ◻
In this section we find exact expansions for the derivatives of the energy. One can see that these align with what we would get by naïvely differentiating the expansion in Lemma 1.
Lemma 7. For \(z\in \mathcal{Z}\) and \(0<\varepsilon<1\) we have \[\partial_\lambda E_\varepsilon[z]=8\pi\mathcal{J}(a)\frac{1}{\lambda^3}+\frac{64\pi}{3c_\gamma}\varepsilon\lambda+\mathcal{O}(\frac{1}{\lambda^4})+\mathcal{O}(\frac{\varepsilon}{\lambda}).\]
Proof. We first note that \[\begin{align} \partial_\lambda E_\varepsilon[z]=&\int_\Sigma \nabla z\cdot\partial_\lambda\nabla z+\varepsilon\Delta z\cdot\partial_\lambda\Delta z\text{d}\Sigma\\ =&\int_{\Sigma\setminus U_r}-\Delta z\cdot\partial_\lambda z+\varepsilon\Delta^2z\cdot\partial_\lambda z\text{d}\Sigma-\int_{\mathbb{D}_r}\Delta z\cdot\partial_\lambda z\text{d}x+\varepsilon\int_{\mathbb{D}_r}(\Delta^g)^2z\cdot\partial_\lambda z\text{d}x_g. \end{align}\] From 27 and 29 we have \(|-\Delta z\cdot\partial_\lambda z+\varepsilon\Delta^2z\cdot\partial_\lambda z|=\mathcal{O}(\frac{1}{\lambda^4})\) on \(\Sigma\setminus U_r\). For the second term we note that \(\Delta\pi_\lambda\cdot\partial_\lambda\pi_\lambda=0\), so using 36 we get \[\Delta z\cdot\partial_\lambda z=\Delta\pi_\lambda\cdot \partial_\lambda j_\lambda^\top+\Delta j_\lambda^\top\cdot \partial_\lambda \pi_\lambda+\mathcal{O}(\frac{1}{\lambda^2}\frac{|x|}{1+\lambda^2|x|^2}).\] By Taylor expanding we obtain \[\begin{align} \Delta\pi_\lambda\cdot \partial_\lambda j_\lambda^\top+\Delta j_\lambda^\top\cdot \partial_\lambda \pi_\lambda =&-\frac{1}{\lambda}(x^a\partial_aj\cdot\pi_\lambda)|\nabla\pi_\lambda|^2\\ =& -\frac{32\lambda}{(1+\lambda^2|x|^2)^3}\bigg((x^1)^2\partial_{x^1y^1}J(0,0)+(x^2)^2\partial_{x^2y^2}J(0,0)\\ &\qquad\qquad\quad+x^1x^2(\partial_{x^1y^2}J(0,0)+\partial_{x^2y^1}J(0,0))\bigg)\\ &+\mathcal{O}(\frac{\lambda|x|^3}{(1+\lambda^2|x|^2)^3}). \end{align}\] Now putting these estimates together along with the fact that we are integrating on a disc, we get \[\begin{align} \begin{aligned}\label{eq:32446132final322} -\int_{\mathbb{D}_r}\Delta z\cdot\partial_\lambda z\text{d}x=&\int_{\mathbb{D}_r}\frac{16\lambda|x|^2}{(1+\lambda^2|x|^2)^3}\mathcal{J}(a)+\mathcal{O}(\frac{\lambda|x|^3}{(1+\lambda^2|x|^2)^3}+\frac{1}{\lambda^2}\frac{|x|}{1+\lambda^2|x|^2})\\ =&8\pi\mathcal{J}(a)\frac{1}{\lambda^3}+\mathcal{O}(\frac{1}{\lambda^4}) \end{aligned} \end{align}\tag{5}\] where \(\mathcal{J}(a)=\partial_{x^1y^1}J_a(0,0)+\partial_{x^2y^2}J_a(0,0)\). For the final term we use 28 and 32 to get on \(\mathbb{D}_r\) \[\begin{align} (\Delta^g)^2z\cdot \partial_\lambda z=&\frac{1}{c_\gamma^2}\Delta^2\pi_\lambda\cdot\partial\pi_\lambda+\mathcal{O}(\frac{\lambda^2|x|}{(1+\lambda^2|x|^2)^{3}})\\ =&\frac{256\lambda^5|x|^2}{(1+\lambda^2|x|^2)^5}+\mathcal{O}(\frac{\lambda^2|x|}{(1+\lambda^2|x|^2)^{3}}). \end{align}\] One can calculate \[\begin{align} \begin{aligned}\label{eq:32446132final323} \int_{\mathbb{D}_r}\frac{|x|^2}{(1+\lambda^2|x|^2)^5}\text{d}x_g =&\frac{\pi}{12}\frac{c_\gamma}{\lambda^4}+\mathcal{O}(\frac{1}{\lambda^6})\\ \int_{\mathbb{D}_r}\frac{\lambda^2|x|}{(1+\lambda^2|x|^2)^{3}}\text{d}x_g=&\mathcal{O}(\frac{1}{\lambda}). \end{aligned} \end{align}\tag{6}\] Combining 5 and 6 completes the proof. ◻
Lemma 8. For \(z\in \mathcal{Z}\) and \(0<\varepsilon<1\) we have \[\nabla_A E_\varepsilon[z]=4\pi\nabla_A\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3})+\mathcal{O}(\varepsilon\lambda).\]
Proof. We first note that \[\begin{align} \nabla_A E_\varepsilon[z]=&\int_\Sigma \nabla z\cdot\nabla_A \nabla z+\varepsilon\Delta z\cdot\nabla_A \Delta z\text{d}\Sigma\\ =&\int_{\Sigma\setminus U_r}-\Delta z\cdot\nabla_A z+\varepsilon\Delta^2 z\cdot\nabla_A z\text{d}\Sigma-\int_{\mathbb{D}_r}\Delta z\cdot\nabla_A z\text{d}x\\ &+\varepsilon\int_{\mathbb{D}_r}(\Delta^g)^2z\cdot \nabla_A z\text{d}x_g. \end{align}\] Using 27 and 31 we have \(|-\Delta z\cdot\nabla_A z+\varepsilon\Delta^2 z\cdot\nabla_A z|=\mathcal{O}(\frac{1}{\lambda^3})\) on \(\Sigma\setminus U_r\). We calculate, using 36 and 37 , \[\begin{align} \Delta z\cdot\nabla_A z=&\Delta\pi_\lambda\cdot\nabla_Aj^\top_\lambda+\Delta j^\top_\lambda\cdot\nabla_A \pi_\lambda+\mathcal{O}(\frac{|x|}{(1+\lambda^2|x|^2)^2}+\frac{1}{\lambda^3})\\ =&|\nabla\pi_\lambda|^2A^m\partial_m j_\lambda\cdot\nabla\pi_\lambda+\mathcal{O}(\frac{|x|}{(1+\lambda^2|x|^2)^2}+\frac{1}{\lambda^3}). \end{align}\] By Taylor expanding we have \[\begin{align} A^m\partial_m j_\lambda\cdot\nabla\pi_\lambda=&\frac{4}{1+\lambda^2|x|^2}\bigg(A^m\partial_mj_\lambda(0)\cdot x\\ &+(x^1)^2(A^1\partial^3_{x^1x^1y^1}J(0,0)+A^2\partial^3_{x^2x^1y^1}J(0,0))\\ &+(x^2)^2(A^1\partial^3_{x^1x^2y^2}J(0,0)+A^2\partial^3_{x^2x^2y^2}J(0,0))+Cx^1x^2\bigg)\\ &+\mathcal{O}(\frac{|x|^3}{1+\lambda^2|x|^2}). \end{align}\] We recall the following lemma.
Lemma 9 (Lemma B.1 [8]). With \(J_a\) and \(\mathcal{J}\) as in the definitions of \(\Sigma\) and \(z\) we have the following. \[-\frac{1}{2}\nabla_A\mathcal{J}(a)=A^i\partial_{x^i}(\partial^2_{x^1y^1}J_a(0,0)+\partial^2_{x^2y^2}J_a(0,0)).\]
This gives us \[\begin{align} -\int_{\mathbb{D}_r}\Delta z\cdot\nabla_A z\text{d}x=&8\lambda^2\nabla_A\mathcal{J}(a)\int_{\mathbb{D}_r}\frac{|x|^2}{(1+\lambda^2|x|^2)^3}+\mathcal{O}(\frac{1}{\lambda^3})\\ =&4\pi\nabla_A\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3}). \end{align}\] For the third term we get, using 25 and 32 , \[\begin{align} (\Delta^g)^2z\cdot \nabla_A z=\frac{1}{c_\gamma^2}\Delta^2\pi_\lambda\cdot(-A^j\nabla_j\pi_\lambda)+\mathcal{O}(\frac{\lambda^3}{(1+\lambda^2|x|^2)^2}). \end{align}\] Then using 22 along with conformality of \(\pi_\lambda\) we get \[\begin{align} \int_{\mathbb{D}_r}\frac{1}{c_\gamma^2}\Delta^2\pi_\lambda\cdot(-A^j\nabla_j\pi_\lambda) \text{d}x_g =&\int_{\mathbb{D}_r}-\frac{1}{c_\gamma^2}\frac{32\lambda^4}{(1+\lambda^2|x|^2)^3}|\nabla\pi_\lambda|^2(A\cdot x)\text{d}x_g=0,\\ \int_{\mathbb{D}_r}\frac{\lambda^3}{(1+\lambda^2|x|^2)^3} \text{d}x_g=&\mathcal{O}(\lambda). \end{align}\] Combining these results completes the proof. ◻
We will now prove a number of bounds on the first and second variation of the \(\varepsilon\)-energy, which we use for the final bound. We shall be concerned with maps with values in \(z^*T\mathbb{S}^2\) almost everywhere, so we define \[\Gamma^2(z)=\{V\in W^{2,2}(\Sigma,\mathbb{R}^3) \text{ s.t. }V(x)\in T_{z(x)}\mathbb{S}^2 \text{ a.e.}\}.\]
Lemma 10. Let \(0\leq \varepsilon,\delta\leq 1\) and \(z\in \mathcal{Z}\) with \(E_\varepsilon[z]\leq\delta\) and \(V\in \Gamma^2(z)\). Then there exists \(C(\Sigma)\) such that \[|\textrm{d}E_\varepsilon(z)[V]|\leq C(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|V\|_z,\] \[|\textrm{d}^2E_\varepsilon[\partial_\lambda z,V]|\leq C\frac{1}{\lambda}(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|V\|_z\] and \[|\textrm{d}^2E_\varepsilon(z)[\nabla_A z,V]|\leq C\lambda(\frac{(\log\lambda)^{1/2}}{\lambda^2}+\varepsilon\lambda^2)\|V\|_z.\]
Proof. We first note that \[\begin{align} \text{d}E_\varepsilon(z)[V]=&\int_\Sigma -\Delta z\cdot V+\varepsilon\Delta^2 z\cdot V\text{d}\Sigma\\ =&\int_{\Sigma\setminus U_r(a)} -\Delta z \cdot V\text{d}\Sigma-\int_{\mathbb{D}_r}\Delta z\cdot V\text{d}x+\varepsilon\int_{\Sigma}\Delta^2 z\cdot V\text{d}\Sigma \end{align}\] For the first term we have that, using 27 and Remark 5, \[\label{eq:32546146132final321} |\int_{\Sigma\setminus U_r(a)} -\Delta z \cdot V|\leq C\frac{1}{\lambda^2}\int_{\Sigma}|V|\leq C\frac{(\log \lambda)^{1/2}}{\lambda^2}\|V\|_z.\tag{7}\] For the second term note that, as \(V\in\Gamma^2(z)\), we only need to consider the part of \(\Delta z\) tangential to \(z\). So using 26 we get \[\begin{align} (\Delta z)^{\top_z}=&(\Delta z)^{\top_\pi}+\mathcal{O}(\frac{\lambda|x|}{(1+\lambda^2|x|^2)^2})\\ =&(\Delta \pi+\mathcal{O}(\frac{1}{(1+\lambda^2|x|^2)^{3/2}}))^{\top_\pi}+\mathcal{O}(\frac{\lambda|x|}{(1+\lambda^2|x|^2)^2})\\ =&\mathcal{O}(\frac{1}{(1+\lambda^2|x|^2)^{3/2}}). \end{align}\] Where we have used that \(|y^{\top_z}-y^{\top_\pi}|\leq |j||y|\leq \frac{|x|}{\lambda}|y|\) for any \(y\in\mathbb{R}^3\). This gives \[\begin{align} \begin{aligned}\label{eq:32546146132final322} |\int_{\mathbb{D}_r}\Delta z\cdot V|\leq&C\int_{\mathbb{D}_r}\frac{1}{(1+\lambda^2|x|^2)^{3/2}}|V|\\ \leq&C\bigg(\int_{\mathbb{D}_r}\frac{1}{\lambda^2(1+\lambda^2|x|^2)}\bigg)^{1/2}\bigg(\int_{\mathbb{D}_r}\rho_z^2|V|\bigg)^{1/2}\\ \leq& C\frac{(\log \lambda)^{1/2}}{\lambda^2}\|V\|_z. \end{aligned} \end{align}\tag{8}\] For the third term we use 33 to get \[\begin{align} \begin{aligned}\label{eq:32546146132final323} |\int_{\Sigma}\Delta^2 z\cdot V|\leq\int_\Sigma\lambda^2\rho_z^2|V|\leq&\bigg(\int_{\Sigma}\lambda^4\rho_z^2\bigg)^{1/2}\bigg(\int_{\Sigma}\rho_z^2 |V|^2\bigg)^{1/2}\\ \leq& C\lambda^2\|V\|_z. \end{aligned} \end{align}\tag{9}\] Combining 7 , 8 and 9 gives the first inequality.
For the second inequality we first note that we have, using Remark 5, 1 , 27 and 29 , \[\begin{align} |\text{d}^2E_\varepsilon(z)[\partial_\lambda z,V]|=&|\int_\Sigma -(\partial_\lambda \Delta z+ |\nabla z|^2\partial_\lambda z)\cdot V+\varepsilon(\partial_\lambda\Delta^2 z-(\Delta^2 z\cdot z)\partial_\lambda z)\cdot V\text{d}\Sigma|\\ \leq&C\frac{(\log \lambda)^{1/2}}{\lambda^3}\|V\|_z-\int_{\mathbb{D}_r}(\partial_\lambda \Delta z+|\nabla z|^2\partial_\lambda z)\cdot V\text{d}x\\ &\quad\quad+\varepsilon\int_{\Sigma}(\partial_\lambda\Delta^2 z-(\Delta^2 z\cdot z)\partial_\lambda z)\cdot V\text{d}\Sigma. \end{align}\] For the second term we note that from 25 , 28 and [eq:32Delta32lambda32z32tangent32bound] we get \[\begin{align} (\partial_\lambda\Delta z+ |\nabla z|^2\partial_\lambda z)^{\top_z}=&(\partial_\lambda\Delta \pi_\lambda +|\nabla \pi_\lambda |^2\partial_\lambda\pi_\lambda)^{\top_z} +\mathcal{O}(\frac{|x|}{\lambda^2}\rho_z^2). \end{align}\] We can then calculate \[(\partial_\lambda\Delta \pi_\lambda +|\nabla \pi_\lambda |^2\partial_\lambda\pi_\lambda )^{\top_z}=-\partial_\lambda(|\nabla \pi_\lambda |^2)\pi_\lambda^{\top_z}=\mathcal{O}(\frac{|x|}{\lambda^2}\rho_z^2).\] Which then gives \[\label{eq:32546146232final322} \begin{align} \int_{\mathbb{D}_r}(\partial_\lambda \Delta z+|\nabla z|^2\partial_\lambda z)\cdot V\text{d}x \leq&\frac{1}{\lambda^2}\int_{\mathbb{D}_r}|x|\rho_z^2|V|\\ \leq&\frac{1}{\lambda^2}(\int_{\mathbb{D}_r}|x|^2\rho_z^2)^{1/2}(\int_{\mathbb{D}_r}\rho_z^2|V|^2)^{1/2} \leq C\frac{(\log\lambda)^{1/2}}{\lambda^3}\|V\|_z. \end{align}\tag{10}\] For the third term we note that using 28 , 33 and 34 we have globally \[|\partial_\lambda(\Delta^g)^2 z-((\Delta^g)^2 z\cdot z)\partial_\lambda z|=\mathcal{O}(\lambda\rho_z^2)\] giving \[\label{eq:32546146232final323} |\int_{\mathbb{D}_r}(\partial_\lambda(\Delta^g)^2 z-((\Delta^g)^2 z\cdot z)\partial_\lambda z)\cdot V\text{d}x_g|\leq C\lambda\int_{\mathbb{D}_r}\rho_z^2|V| \leq C \lambda\|V\|_z .\tag{11}\] Combining 10 and 11 gives the second inequality.
Now for the final inequality we have, using 1 , 27 and Remark 5, \[\begin{align} |\text{d}^2E_\varepsilon(z)[\nabla_A z,V]|\leq&C \frac{(\log\lambda)^{1/2}}{\lambda}\|V\|_z+|\int_{\mathbb{D}_r} \nabla \nabla_A z\cdot \nabla V-(|\nabla z|^2\nabla_A z)\cdot V\text{d}x|\\ &+\varepsilon|\int_\Sigma(\Delta^2 \nabla_Az -(\Delta^2 z\cdot z) \nabla_Az)\cdot V\text{d}\Sigma|.\\ \end{align}\] From 25 and 30 we have on \(\mathbb{D}_r\) \[\begin{align} \nabla \nabla_A z=& A^j\partial_j\nabla \pi_\lambda+\mathcal{O}(\frac{1}{(1+\lambda^2|x|^2)^{1/2}})\\ |\nabla z|^2\nabla_A z =& |\nabla\pi_\lambda|^2A^j\partial_j \pi_\lambda+\mathcal{O}(\frac{\lambda}{(1+\lambda^2|x|^2)^2}). \end{align}\] We calculate \[\int_{\mathbb{D}_r}\frac{1}{(1+\lambda^2|x|^2)^{1/2}}|\nabla V|+\frac{\lambda}{(1+\lambda^2|x|^2)^2}|V|\leq C\frac{(\log\lambda)^{1/2}}{\lambda}\|V\|_z.\] We also find that \[(A^j\partial_j\Delta \pi_\lambda+ |\nabla \pi_\lambda|^2A^j\partial_j \pi_\lambda)^{\top_z}=-(A^j\partial_j|\nabla \pi_\lambda|^2)(\pi_\lambda)^{\top_z}=\mathcal{O}(\frac{|x|^2\lambda^3}{(1+\lambda^2|x|^2)^{3}})\] which gives, further using Remark 5, \[\begin{align} \begin{aligned}\label{eq:32546146332final322} |\int_{\mathbb{D}_r} A^j\partial_j\nabla \pi_\lambda\cdot \nabla V-(|\nabla \pi_\lambda|^2A^j\partial_j \pi_\lambda)\cdot V|\leq&\int_{\partial\mathbb{D}_r}|A^j\partial_j\nabla \pi_\lambda|\cdot|V|\\&+ |\int_{\mathbb{D}_r}(A^j\partial_j\Delta \pi_\lambda+ |\nabla \pi_\lambda|^2A^j\partial_j \pi_\lambda)\cdot V|\\ \leq &C\frac{1}{\lambda}\int_{\partial\mathbb{D}_r}|V|+C\int_{\mathbb{D}_r}\frac{|x|^2\lambda^2}{(1+\lambda^2|x|^2)^{2}}\rho_z|V|\\ \leq & \frac{(\log\lambda)^{1/2}}{\lambda}\|V\|_z. \end{aligned} \end{align}\tag{12}\] We also have from 33 and 35 the estimate that globally \[\Delta^2\nabla_Az+(\Delta^2 z\cdot z) \nabla_A z=\mathcal{O}(\lambda^3\rho_z^2).\] Which gives \[\label{eq:32546146332final323} |\int_\Sigma (\Delta^2\nabla_Az+(\Delta^2 z\cdot z) \nabla_A z)\cdot V|\leq C\lambda^3\int_\Sigma\rho_z^2|V|\leq C\lambda^3\|V\|_z.\tag{13}\] Combining 12 and 13 completes the proof. ◻
Lemma 11. Given \(u\in W^{2,\infty}(\Sigma,\mathbb{S}^2)\) and \(z\in\mathcal{Z}\) with \(\|u-z\|_{L^\infty}\leq \frac{1}{2}\). Set \(w=u-z\), \(u_t=P(z+tw)\), \(W_t=\textrm{d}P(z+tw)[w]\) and \(W=W_0\). Then there exists some constant \(C=C(\Sigma)\) such that \[|\textrm{d}^2E_\varepsilon(z)[W,W]-\textrm{d}^2E_\varepsilon(u_t)[W_t,W_t]|\leq C(\|w\|_{L^\infty}\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^{\infty}})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2)\] and \[|\int_\Sigma \textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)[\partial_tW_t]]|\leq C(\|w\|_{L^\infty}\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2).\]
Proof. We can calculate \(w^{\top_z}=(w\cdot z)z=-\frac{1}{2}|w|^2z\) and \(W=w+\frac{1}{2}|w|^2z\). We then have \[\nabla W=\nabla w+(w\cdot\nabla w)z+\frac{1}{2}|w|^2\nabla z.\] This gives, using the uniform \(L^\infty\) bound on \(w\), that
\[\begin{align} \label{lkgimeja} C_1\|w\|_{L^\infty}\leq\|W\|_{L^\infty}\leq C_2\|w\|_{L^\infty}, \quad\quad C_3\|w\|_z^2\leq\|W\|_z^2\leq C_4\|w\|_z^2. \end{align}\tag{14}\] We then have the bounds on \(W_t\) from the definitions, using that \(|\nabla z|\leq C\rho_z\) from 25 and 27 , \[\label{eq:32first32W95t32bounds} \begin{align} |W_t|\leq& C|w|,\\ |\nabla W_t|\leq& C(|\nabla w|+|w|\rho_z),\\ \end{align} \begin{align} |W_t-W|\leq& C|w|^2,\\ |\nabla (W_t- W)|\leq& C(|w|\cdot|\nabla w|+|w|^2\rho_z). \end{align}\tag{15}\] Also note that \[\begin{align} \Delta W_t=& dP(z+tw)[\Delta w]+2d^2P(z+tw)[\nabla w,\nabla z+t\nabla w]\\ &+d^2P(z+tw)[w,\Delta z+t\Delta w] +d^3P(z+tw)[w,\nabla z+t\nabla w,\nabla z+t\nabla w]. \end{align}\] Which gives us the bounds, also using 26 and 27 , \[\begin{align} \begin{aligned}\label{eq:32Delta32W95t32bounds} |\Delta W_t|\leq& C(|\Delta w|+|\nabla w|\rho_z+|\nabla w|^2+|w|\rho_z^2),\\ |\Delta (W_t-W)|\leq & C(|w|\cdot|\Delta w|+|w|\cdot|\nabla w|\rho_z+|\nabla w|^2+|w|^2\rho_z^2). \end{aligned} \end{align}\tag{16}\] We also note that \[\Delta u_t=\text{d}P(z+tw)[\Delta z+t\Delta w]+\text{d}P^2(z+tw)[\nabla z+t\nabla w,\nabla z+t\nabla w]\] giving the bounds on \(u_t\), using the fact that \(z=u_0\), \[\label{eq:32u95t32bounds} \begin{align} |u_t|\leq& C\\ |\nabla u_t|\leq& C(|\nabla w|+\rho_z)\\ |\Delta u_t|\leq &C(|\nabla w|^2+|\nabla w|\rho_z\\&+\rho_z^2+|\Delta w|)\\ \end{align}\begin{align} |u_t-z|\leq& C|w|\\ |\nabla (u_t-z)|\leq& C(|\nabla w|+|w|\rho_z)\\ |\Delta (u_t-z)|\leq& C(|\nabla w|^2+|\nabla w|\rho_z\\&+|w|\rho_z^2+|\Delta w|).\\ \end{align}\tag{17}\] We can then calculate, using 1 , \[\begin{align} &\text{d}^2E_\varepsilon(z)[W,W]-\text{d}^2E_\varepsilon(u_t)[W_t,W_t]=\\ &\int_\Sigma \nabla (W-W_t)\cdot\nabla(W+W_t) -|\nabla z|^2(|W|^2-|W_t|^2) -(|\nabla z|^2-|\nabla u_t|^2)|W_t|^2\\&+\varepsilon\Delta (W-W_t)\cdot\Delta(W+W_t) -\varepsilon\Delta^2z\cdot(z|W|^2-u_t|W_t|^2)-\varepsilon\Delta(z-u_t)\cdot\Delta(u_t|W_t|^2). \end{align}\] So then using 15 , 16 and 17 and working through the terms we get \[\begin{align} |\int_\Sigma \nabla (W-W_t)\cdot\nabla(W+W_t)| \leq& C\int_\Sigma (|w|\cdot|\nabla w|+|w|^2\rho_z)(|\nabla w|+|w|\rho_z)\\ \leq& C\|w\|_{L^\infty}\|w\|_z^2,\\ |\int_\Sigma |\nabla z|^2(|W|^2-|W_t|^2)|=&|\int_\Sigma |\nabla z|^2 (W-W_t)\cdot(W+W_t)|\\ \leq&C\int_\Sigma |w|^3\rho_z^2\\ \leq& C\|w\|_{L^\infty}\|w\|_z^2,\\ |\int_\Sigma(|\nabla z|^2-|\nabla u_t|^2)|W_t|^2|=& |\int_\Sigma(\nabla z-\nabla u_t)\cdot(\nabla z+\nabla u_t)|W_t|^2|\\ \leq&C\int_\Sigma(|\nabla w|+|w|\rho_z)(|\nabla w|+\rho_z)|w|^2\\ \leq& C\|w\|_{L^\infty}\|w\|_z^2,\\ |\int_\Sigma \Delta (W-W_t)\cdot\Delta (W+W_t)|\leq& C\int_\Sigma(|\Delta w|+|\nabla w|\rho_z+|\nabla w|^2+|w|\rho_z^2)\\ &\quad\cdot(|w|\cdot|\Delta w|+|w|\cdot|\nabla w|\rho_z+|\nabla w|^2+|w|^2\rho_z^2)\\ \leq&C((\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\|\Delta w\|_{L^2}^2). \end{align}\] For the next term note that \[z|W|^2-u_t|W_t|^2=(z-u_t)|W|^2+u_t(W-W_t)\cdot (W+W_t)\] so using 15 , 17 and 33 \[| \int_\Sigma \Delta^2z\cdot(z|W|^2-u_t|W_t|^2)|\leq C\int_\Sigma\lambda^2\rho_z^2|w|^3\leq C\lambda^2\|w\|^2_z.\] For the final term we calculate, using 15 , 16 and 17 , that \[\begin{align} \Delta(u_t|W_t|^2)=&(\Delta u_t)|W_t|^2+4\nabla u_t\cdot\nabla W_t\cdot W_t+2u_t(\Delta W_t\cdot W_t+|\nabla W_t|^2)\\ \leq& C(|w|^2\rho_z^2+|w|\cdot|\nabla w|\rho_z+|\nabla w|^2+|w|\cdot|\Delta w|). \end{align}\] So using 17 \[\begin{align} &|\int_\Sigma \Delta(z-u_t)\cdot\Delta(u_t|W_t|^2)|\\ &\leq C\int_\Sigma(|\nabla w|^2+|\nabla w|\rho_z+|w|\rho_z^2+|\Delta w|)\\ &\quad\quad\cdot(|w|^2\rho_z^2+|w|\cdot|\nabla w|\rho_z+|\nabla w|^2+|w|\cdot|\Delta w|)\\ &\leq C((\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\|\Delta w\|^2_{L^2}). \end{align}\] Combining these estimates gives the first equation.
For the second equation we first note that \[\begin{align} W_t=\partial_tu_t=&\partial_t\frac{z+tw}{|z+tw|}\\ =&\frac{w}{|z+tw|}-\frac{w\cdot(z+tw)}{|z+tw|^3}(z+tw)\\ =&\frac{1}{|z+tw|}\text{d}P(u_t)[w]. \end{align}\] We then find that \[\begin{align} \text{d}P(u_t)[\partial_t W_t] =&-2\frac{1}{|z+tw|}(w\cdot u_t)W_t\\ =&\frac{1}{|z+tw|}(|w|^2-2w\cdot (u_t-z))W_t \end{align}\] using \(w\cdot(w+2z)=0\). Now directly differentiating and using [eq:32w32and32W32equiv46], 15 , 16 and 17 gives \[\begin{align} \begin{aligned}\label{eq:32dP32of32partial95t32W95t32bounds} |\text{d}P(u_t)[\partial_tW_t]|\leq& C |w|^3,\\ |\nabla\text{d}P(u_t)[\partial_tW_t]|\leq&C(|w|^2|\nabla w|+|w|^3\rho_z),\\ |\Delta\text{d}P(u_t)[\partial_tW_t]|\leq&C(|w|^2|\Delta w|+|w|\cdot|\nabla w|^2+|w|^2|\nabla w|\rho_z+|w|^3\rho_z^2). \end{aligned} \end{align}\tag{18}\] This then gives, using 1 , 17 and 18 , \[\begin{align} |\int_\Sigma \text{d}E_\varepsilon(u_t)[\text{d}P(u_t)[\partial_tW_t]]|\leq &C\int_\Sigma(|\nabla w|+\rho_z)(|w|^2|\nabla w|+|w|^3\rho_z)\\ &+C\varepsilon\int_\Sigma(|\nabla w|^2+|\nabla w|\rho_z+\rho_z^2+|\Delta w|)\\ &\quad\quad\cdot(|w|^2|\Delta w|+|w|\cdot|\nabla w|^2+|w|^2|\nabla w|\rho_z+|w|^3\rho_z^2)\\ \leq&C(\|w\|_{L^\infty}\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2) \end{align}\] completing the proof. ◻
Lemma 12. Given \(u\in W^{2,\infty}(\Sigma,\mathbb{S}^2)\) and \(z\in\mathcal{Z}\) with \(\|u-z\|_{L^\infty}\leq \frac{1}{2}\). Set \(w=u-z\), \(u_t=P(z+tw)\), \(W_t=\textrm{d}P(z+tw)[w]\) and \(W=W_0\). Also given \(T\in\Gamma^2(z)\) set \(T_t=\textrm{d}P(u_t)[T]\). Then there exists some constant \(C=C(\Sigma)\) such that
If \(T=\partial_\lambda z\) then, \[|\textrm{d}^2E_\varepsilon(z)[T,W]-\textrm{d}^2E_\varepsilon(z)[T_t,W_t]|\leq C\frac{1}{\lambda}(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2)\] and \[|\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)[\partial_t T_t^\lambda]]|\leq C\frac{1}{\lambda}(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2).\]
If \(T=\nabla_Az\) then, \[|\textrm{d}^2E_\varepsilon(z)[T,W]-\textrm{d}^2E_\varepsilon(z)[T_t,W_t]|\leq C\lambda(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2)\] and \[|\textrm{d}E_\varepsilon(u_t)[\textrm{d}P(u_t)[\partial_t T_t^A]]|\leq C\lambda(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2).\]
Proof. Note that we have, using 1 , \[\begin{align} \begin{aligned}\label{eq:32second32variation32T32expansion} \text{d}^2E_\varepsilon(z)[T,W]-\text{d}^2E_\varepsilon(z)[T_t,W_t]=&\int_\Sigma\nabla W\cdot\nabla(T-T_t)+\nabla(W-W_t)\cdot\nabla T_t\\ &\quad-(|\nabla z|^2-|\nabla u_t|^2)(T\cdot W)-|\nabla u_t|^2(T\cdot W-T_t\cdot W_t)\\ &+\varepsilon\int_\Sigma\Delta W\cdot\Delta(T-T_t)+\Delta(W-W_t)\cdot\Delta T_t\\ &\quad-\Delta^2z\cdot(z(T\cdot W)-u_t(T_t\cdot W_t))\\ &\quad-\Delta(z-u_t)\cdot\Delta(u_t(T_t\cdot W_t)). \end{aligned} \end{align}\tag{19}\] Also note that \[\begin{align} \nabla T_t=&\text{d}P(u_t)[\nabla T]+\text{d}^2P(u_t)[T,\nabla u_t],\\ \Delta T_t=&\text{d}P(u_t)[\Delta T]+\text{d}^2P(u_t)(2[\nabla T,\nabla u_t]+[T,\Delta u_t])+\text{d}^2P(u_t)[T,\nabla u_t,\nabla u_t]. \end{align}\] This gives, using 17 , \[\begin{align}\label{eq:32Initial32T32bounds} |T_t|\leq& |T|,\\ |\nabla T_t|\leq &C\big(|T|(\rho_z+|\nabla w |)+|\nabla T|\big),\\ |\Delta T_t|\leq &C\big(|T|(\rho_z^2+|\nabla w|\rho_z+|\nabla w|^2+|\Delta w|)\\ &\quad\quad+|\nabla T|(\rho_z+|\nabla w |)+|\Delta T|\big),\\ |T_t-T|\leq& C|T|\cdot|w|,\\ |\nabla(T_t-T)|\leq& C\big(|T|(|w|\rho_z+|\nabla w |)+|\nabla T|\cdot|w|\big),\\ |\Delta(T_t-T)|\leq& C\big(|T|(|w|\rho_z^2+|\nabla w|\rho_z+|\nabla w|^2+|\Delta w|)\\ &\quad\quad+|\nabla T|(|w|\rho_z+|\nabla w |)+|\Delta T|\cdot|w|\big). \end{align}\tag{20}\] In the case of \(T=\partial_\lambda z\) we have the bounds from 28 and 29 , \[\begin{align} |T|\leq& C\frac{1}{\lambda}\\ |\nabla T|\leq&C\frac{1}{\lambda}\rho_z\\ |\Delta T|\leq&C \frac{1}{\lambda}\rho_z^2 \end{align}\] this then gives the bounds using 20 \[\begin{align} \begin{aligned}\label{eq:32T94lambda32bounds} |T_t|\leq&C\frac{1}{\lambda},\\ |\nabla T_t|\leq &C\frac{1}{\lambda}(\rho_z+|\nabla w |),\\ |\Delta T_t|\leq &C\frac{1}{\lambda}(\rho_z^2+\rho_z|\nabla w|\\&+|\nabla w|^2+|\Delta w|), \end{aligned}\begin{align} |T_t-T|\leq& C\frac{1}{\lambda}|w|,\\ |\nabla(T_t-T)|\leq& C\frac{1}{\lambda}(\rho_z|w|+|\nabla w |),\\ |\Delta(T_t-T)|\leq& C\frac{1}{\lambda}(\rho_z^2|w|+\rho_z|\nabla w|\\&+|\nabla w|^2+|\Delta w|). \end{align} \end{align}\tag{21}\]
Inserting these bounds into 19 as well as using bounds from 15 , 16 , 17 , 21 and 33 one can obtain the first inequality.
To get the second inequality we note that for \(T\in \Gamma^2(z)\) we have \[\begin{align} \text{d}P(u_t)[\partial_t T_t] =&\text{d}P(u_t)[-(T\cdot W_t)u_t-(T\cdot u_t)W_t]\\ =&-(T\cdot u_t)W_t\\ =&(T\cdot(z-u_t))W_t \end{align}\] using \(T\cdot z=0\). By differentiating this form and appealing to 15 , 16 , 17 and 21 we get the bounds \[\begin{align} |\text{d}P(u_t)[\partial_t T_t]|\leq&C|T|\cdot|w|^2\\ |\nabla \text{d}P(u_t)[\partial_t T_t]|\leq&C\big(|T|(|w|^2\rho_z+|w|\cdot|\nabla w|)+|\nabla T|\cdot|w|^2\big)\\ |\Delta \text{d}P(u_t)[\partial_t T_t]|\leq & C\big(|T|(|\nabla w|^2+|w|\cdot|\nabla w|\rho_z+|w|^2\rho_z^2+|w|\cdot|\Delta w|)\\ &+|\nabla T|(|w|^2\rho_z+|w|\cdot|\nabla w|)+|\Delta T|\cdot|w|^2\big). \end{align}\] In the case of \(T=\partial_\lambda z\), this gives \[\begin{align} |\text{d}P(u_t)[\partial_t T_t]|\leq&C\frac{1}{\lambda}|w|^2,\\ |\nabla \text{d}P(u_t)[\partial_t T_t]|\leq&C\frac{1}{\lambda}(\rho_z|w|^2+|w|\cdot|\nabla w|),\\ |\Delta \text{d}P(u_t)[\partial_t T_t]|\leq & C\frac{1}{\lambda}(|\nabla w|^2+\rho_z|w|\cdot|\nabla w|+\rho_z^2|w|^2+|w|\cdot|\Delta w|). \end{align}\] From the above, 1 and 17 we obtain \[\begin{align} \text{d}E_\varepsilon(u_t)[\text{d}P(u_t)[\partial_t T_t^\lambda]]\leq & \int_\Sigma \nabla u_t\cdot\nabla \text{d}P(u_t)[\partial_t T_t]+\varepsilon \Delta u_t\cdot\Delta \text{d}P(u_t)[\partial_t T_t]\\ \leq&C\frac{1}{\lambda}(\|w\|_z^2+\varepsilon(\lambda^2+\|\nabla w\|^2_{L^\infty})\|w\|_z^2+\varepsilon\|\Delta w\|_{L^2}^2) \end{align}\] giving the second inequality.
In the case of \(T=\nabla_A z\) we instead have the bounds from 30 and 31 \[\begin{align} |T|\leq&C\lambda\\ |\nabla T|\leq&C{\lambda}\rho_z\\ |\Delta T|\leq & C\lambda\rho_z^2 \end{align}\] so the proof for the final two inequalities is identical. ◻
In general we know that close enough to the bubble point \(z\) will look like \(\pi_\lambda\) plus some lower order remainder term for \(\lambda\) large enough. We would like to show this quantitatively. We start by noting the following exact derivatives of \(\pi_\lambda\) \[\begin{align} \begin{aligned}\label{eq:32exact32pi32derivatives} \pi_\lambda=&(\frac{2\lambda x}{1+\lambda^2|x|^2},\frac{1-\lambda^2|x|^2}{1+\lambda^2|x|^2}),\\ \nabla\pi_\lambda=&\frac{2\lambda }{(1+\lambda^2|x|^2)^2} \begin{pmatrix} 1-\lambda^2x_1^2+\lambda^2x_2^2 & -2\lambda^2x_1x_2 & -2\lambda x_1\\ -2\lambda^2x_1x_2 & 1+\lambda^2x_1^2-\lambda^2x_2^2 & -2\lambda x_2 \end{pmatrix},\\ \Delta\pi_\lambda=&\frac{-8\lambda^2}{(1+\lambda^2|x|^2)^2}\pi_\lambda,\\ \nabla_i\Delta\pi_\lambda=&\frac{32\lambda^4x_i}{(1+\lambda^2|x|^2)^3}\pi_\lambda-\frac{8\lambda^2}{(1+\lambda^2|x|^2)^2}\nabla_i\pi_\lambda,\\ \Delta^2\pi_\lambda=&\frac{96\lambda^4-160\lambda^6|x|^2}{(1+\lambda^2|x|^2)^4}\pi_\lambda+\frac{64\lambda^4x_i}{(1+\lambda^2|x|^2)^3}\nabla_i\pi_\lambda.\\ \end{aligned} \end{align}\tag{22}\] It is easy to see that for any \(m\geq 1\) and \(x\in\mathbb{D}_r\) we have the following bounds \[\begin{align} \begin{aligned}\label{eq:32naive32pi32derivative32bounds} |\nabla^m\pi_\lambda|\leq& C(m)\frac{\lambda^m}{(1+\lambda^2|x|^2)^{(m+1)/2}},\\ |\nabla^m\Delta\pi_\lambda|\leq &C(m)\frac{\lambda^{m+2}}{(1+\lambda^2|x|^2)^{m/2+2}}. \end{aligned} \end{align}\tag{23}\] We also have in \(\mathbb{D}_r\) for \(m\geq 1\) \[\begin{align} \begin{aligned}\label{eq:32naive32j32derivative32bounds} |j_\lambda|\leq &C\frac{|x|}{\lambda},\\ |\nabla^mj_\lambda|\leq& C(m)\frac{1}{\lambda}. \end{aligned} \end{align}\tag{24}\] In \(\mathbb{D}_r\) we have \(z=P(\pi_\lambda+j_\lambda)\). We can differentiate this equation to get \[\begin{align} \nabla z=& \text{d}P(\pi_\lambda+j_\lambda)[\nabla\pi_\lambda+\nabla j_\lambda]\\ =&\nabla\pi_\lambda+\text{d}P(\pi_\lambda+j_\lambda)[\nabla j_\lambda]+(\text{d}P(\pi_\lambda+j_\lambda)-\text{d}P(\pi_\lambda))[\nabla\pi_\lambda] \end{align}\] and in general we see that \(\nabla^kz=\nabla^k\pi_\lambda+R_k\) where \(R_k\) is a remainder consisting of terms of the following forms
\((\text{d}^tP(\pi_\lambda+j_\lambda)-\text{d}^tP(\pi_\lambda))[\nabla^{a_1}\pi_\lambda,\dots,\nabla^{a_t}\pi_\lambda]\) where all the \(a_i\) are strictly positive integers and \(a_1+...+a_t=k\)
\(\text{d}^tP(\pi_\lambda+j_\lambda)[\nabla^{b_1}j_\lambda,\dots,\nabla^{b_u}j_\lambda,\nabla^{c_1}\pi_\lambda,\dots,\nabla^{c_v}\pi_\lambda]\) where \(u\), the \(b_i\) and the \(c_i\) are strictly positive integers, \(v\) is a non negative integer, \(u+v=t\) and \(\Sigma_ib_i+\Sigma_jc_j=k\).
Applying 23 and 24 to these remainders we obtain the bounds in \(\mathbb{D}_r\) \[\begin{align} \begin{aligned}\label{eq:32z32remainders} z=&\pi_\lambda+\mathcal{O}(\frac{|x|}{\lambda}),\\ \nabla^k z=&\nabla^k\pi+\mathcal{O}(\frac{\lambda^{k-2}}{(1+\lambda^2|x|^2)^{(k-1)/2}})\quad\text{ for }k\geq 1. \end{aligned} \end{align}\tag{25}\] For Laplacian terms we can get a slightly stronger bound. This uses the fact that \(\Delta j_\lambda=0\). We obtain in \(\mathbb{D}_r\) \[\label{eq:32Delta32z32in32the32ball} \nabla^k\Delta z=\nabla^k\Delta \pi_\lambda+\mathcal{O}(\frac{\lambda^{k}}{(1+\lambda^2|x|^2)^{k/2+1}})\quad\text{ for }k\geq 0.\tag{26}\]
Using the nature of \(z\) on \(\mathbb{D}_{2r}\setminus\mathbb{D}_r\) and on \(\Sigma\setminus U_{2r}(a)\), it is easy to see that \[\begin{align} \begin{aligned}\label{eq:32z32bounds32away32from32D95r} |z|=&\mathcal{O}(1),\\ |\nabla^k z|=&\mathcal{O}(\frac{1}{\lambda})\quad\text{ for }k\geq 1,\\ |\nabla^k\Delta z|=&\mathcal{O}(\frac{1}{\lambda^2})\quad\text{ for }k\geq 0 \end{aligned} \end{align}\tag{27}\] on \(\Sigma\setminus U_{r}(a)\).
We first note that \[\begin{align} \partial_\lambda\pi_\lambda= &\frac{1}{\lambda}(x\cdot\nabla)\pi_\lambda,\\ \partial_\lambda j_\lambda =& -\frac{1}{\lambda}j. \end{align}\] These gives us the following bounds for \(m\geq1\) \[\begin{align} |\nabla^m\partial_\lambda\pi_\lambda|\leq& C(m)\frac{\lambda^{m-1}}{(1+\lambda^2|x|^2)^{(m+1)/2}},\\ |\partial_\lambda j_\lambda|\leq & C \frac{|x|}{\lambda^2},\\ |\nabla^m\partial_\lambda j_\lambda|\leq& C(m)\frac{1}{\lambda^2}. \end{align}\] Then we note that \(\partial_\lambda \nabla^kz=\partial_\lambda \nabla^k\pi_\lambda+\partial_\lambda R_k\) so by differentiating our remainders from before one can obtain the bounds in \(\mathbb{D}_r\) \[\begin{align} \begin{aligned}\label{eq:32lambda32z32remainders} \partial_\lambda z=&\partial_\lambda\pi_\lambda+\mathcal{O}(\frac{|x|}{\lambda^2}),\\ \partial_\lambda \nabla^k z=& \partial_\lambda \nabla^k \pi_\lambda+\mathcal{O}(\frac{\lambda^{k-3}}{(1+\lambda^2|x|^2)^{(k-1)/2}})\quad\text{ for }k\geq 1,\\ \partial_\lambda\nabla^k\Delta z=&\partial_\lambda\nabla^k\Delta \pi_\lambda+\mathcal{O}(\frac{\lambda^{k-1}}{(1+\lambda^2|x|^2)^{k/2+1}})\quad\text{ for }k\geq 0. \end{aligned} \end{align}\tag{28}\]
Also by a similar analysis to before of \(z\) on the annulus \(\mathbb{D}_{2r}\setminus\mathbb{D}_r\) and on the rest of \(\Sigma\), we obtain on \(\Sigma\setminus U_r\) \[\begin{align} \begin{aligned}\label{eq:32lambda32z32bounds32away32from32D95r} |\partial_\lambda \nabla^k z|=&\mathcal{O}(\frac{1}{\lambda^2})\quad\text{ for }k\geq 0,\\ |\partial_\lambda \nabla^k\Delta z|=&\mathcal{O}(\frac{1}{\lambda^3})\quad\text{ for }k\geq 0. \end{aligned} \end{align}\tag{29}\]
For the \(\nabla_A\) derivatives we need to take more care. In the flat case we are working on a homogeneous domain so our choice of base point does not change anything and all derivatives are 0. Now in the hyperbolic case given \(A\in T_a\Sigma\) we can set \(a_s=\text{Exp}_a^\Sigma(sA)\), some path in \(\Sigma\) with derivative \(A\) at \(a\). Then \[\nabla_Az_{\lambda,a}=\frac{\partial}{\partial s}z_{\lambda,a_s}\bigg\vert_{s=0}.\] Then by explicit calculation we have inside \(\mathbb{D}_r\) that, as in [8], \[\nabla_A z = - A^i\partial_iz-|x|^2A^i\partial_iz+2(A\cdot x)x^i\partial_i z.\] In general we can write for \(k\geq 1\) that \(\nabla^k\nabla_A z=-\nabla^kA^i\partial_iz+R_k\) where we can bound \(R_k\) by \[\label{eq:32nabla32A32z32remainders} |R_k|\leq C(|x|^2|\nabla^{k+1}z|+|x|\cdot|\nabla^kz|+|\nabla^{k-1}z|)\leq C \frac{\lambda^{k-1}}{(1+\lambda^2|x|^2)^{k/2}}.\tag{30}\] It is easy to see that on \(\Sigma\setminus U_r\) we get for all \(k\geq 0\) \[\label{eq:32nabla32A32z32away32from32D95r} |\nabla^k\nabla_Az|\leq \frac{1}{\lambda}.\tag{31}\]
For higher order terms in the hyperbolic case we also get terms arising from the metric. In the hyperbolic case inside \(\mathbb{D}_r\) we have, using 26 , \[\begin{align} \begin{aligned}\label{eq:32Delta32squared32z32metric32bound} (\Delta^g)^2z=&\frac{(1-|x|^2)^2}{4}\Delta(\frac{(1-|x|^2)^2}{4}\Delta z)\\ =&(\frac{(1-|x|^2)^2}{4})^2\Delta^2z+\mathcal{O}(|x|)\nabla\Delta z+\mathcal{O}(1)\Delta z\\ =&\frac{1}{c_\gamma^2}\Delta^2\pi_\lambda+\mathcal{O}(\frac{\lambda^2}{(1+\lambda^2|x|^2)^2}). \end{aligned} \end{align}\tag{32}\] The final bound also holds in the flat case. Using 27 , this gives in particular the global bound \[\label{eq:32Delta94232z32global32bound} |\Delta^2 z|\leq C\lambda^2\rho_z^2.\tag{33}\] Differentiating through \(\lambda\) and using 28 also gives the result in \(\mathbb{D}_r\) \[\label{eq:32lambda32delta94232z32global32bound} \partial_\lambda(\Delta^g)^2 z=\frac{1}{c_\gamma^2}\partial_\lambda\Delta^2\pi_\lambda+\mathcal{O}(\frac{\lambda}{(1+\lambda^2|x|^2)^2})\tag{34}\] which clearly also holds in the flat case. This and 29 then give the global bound \[|\partial_\lambda\Delta^2z|\leq C\lambda\rho_z^2.\] For \(\nabla_A\), we get in \(\mathbb{D}_r\) that \[\begin{align} (\Delta^g)^2\nabla_Az=& -A^i\Delta^2 \partial_iz+\mathcal{O}(|x|^2)\nabla\Delta^2z\\ &\quad+\mathcal{O}(|x|)\nabla^2\Delta z+\mathcal{O}(1)\nabla^3 z+\mathcal{O}(1)\nabla^2z+\mathcal{O}(1)\nabla z\\ =&-A^i\Delta^2 \partial_iz+\mathcal{O}(\frac{\lambda^3}{(1+\lambda^2|x|^2)^2}). \end{align}\] Which with 31 gives us a global bound of \[\label{eq:32delta94232nabla32A32z32global32bound} |\Delta^2\nabla_A z |\leq C\lambda^3\rho_z^2.\tag{35}\]
For the lower order terms we will need some sharper estimates in order to obtain the dependence on \(\mathcal{J}\). Inside of \(\mathbb{D}_r\) one can view \(z\) as \(z=\pi_\lambda+j_\lambda^\top+K_\lambda\) where \[j_\lambda^\top=\text{d}P(\pi_\lambda)[j_\lambda]\] is the projection of \(j_\lambda\) onto the plane tangent to \(\pi_\lambda\) and \[K_\lambda=z-\pi_\lambda-j_\lambda^\top=\int^1_0\text{d}P(\pi_\lambda+tj_\lambda)[j_\lambda]-\text{d}P(\pi_\lambda)[j_\lambda] \text{d}t\] We then have the following extra bounds on \(j_\lambda\) and \(K_\lambda\) in \(\mathbb{D}_r\) as in [8]. \[\label{eq:32extra32j32and32K32bounds} \begin{align} |j_\lambda|=&\mathcal{O}(\frac{|x|}{\lambda}) ,\\ |\nabla j_\lambda^\top|=&\mathcal{O}(\frac{1}{\lambda}) ,\\ |\Delta j_\lambda^\top|=&\mathcal{O}(\frac{1}{(1+\lambda^2|x|^2)}) ,\\ |\nabla^2 j_\lambda^\top|=&\mathcal{O}(\frac{1}{\lambda}+\frac{1}{(1+\lambda^2|x|^2)}) ,\\ |\partial_\lambda j_\lambda^\top|=&\mathcal{O}(\frac{|x|}{\lambda^2}) ,\\ \end{align} \quad\quad\quad\quad \begin{align} |K_\lambda|=&\mathcal{O}(\frac{|x|^2}{\lambda^2}),\\ |\nabla K_\lambda|=&\mathcal{O}(\frac{1}{\lambda^2}),\\ |\Delta K_\lambda|=&\mathcal{O}(\frac{1}{\lambda^2}),\\ |\nabla^2 K_\lambda|=&\mathcal{O}(\frac{1}{\lambda^2}),\\ |\partial_\lambda K_\lambda|=&\mathcal{O}(\frac{|x|^2}{\lambda^3}).\\ \end{align}\tag{36}\] We also note the additional bounds \[\begin{align} \begin{aligned}\label{eq:32sharper32j32and32K32bounds} |(\Delta K_\lambda)^\top|=&\mathcal{O}(\frac{1}{\lambda}\frac{|x|}{1+\lambda^2|x|^2}),\\ |\Delta j_\lambda^\top\cdot \nabla_Aj^\top_\lambda| &=\mathcal{O}(\frac{|x|}{(1+\lambda^2|x|^2)^2}). \end{aligned} \end{align}\tag{37}\]
We also note that by expanding out and using the exact form of \(\text{d}^2P_{\mathbb{S}^2}\) we obtain \[\label{enbompxr} ( \Delta\partial_\lambda z)^{\top_z}= (\Delta\partial_\lambda \pi_\lambda)^{\top_z}+\mathcal{O}(\frac{|x|}{\lambda^2}\rho_z^2).\tag{38}\]
To finish the proof of Lemma 1 we need to expand the Dirichlet energy along \(\mathcal{Z}\). For \(z\in\mathcal{Z}\) we claim \[E[z]=\frac{1}{2}\int_\Sigma|\nabla z|^2=4\pi-4\pi\mathcal{J}(a)\frac{1}{\lambda^2}+\mathcal{O}(\frac{1}{\lambda^3}).\] Using \(\eqref{eq:32naive32pi32derivative32bounds}\) and \(\eqref{eq:32naive32j32derivative32bounds}\) one can show that \[\begin{align} \int_{\mathbb{D}_{2r}}|\nabla z|^2=&\int_{\mathbb{D}_{2r}}|\nabla \pi|^2+2\Delta \pi\cdot j+2\nabla \pi\cdot \nabla j+|\nabla j|^2+\mathcal{O}(\frac{1}{\lambda^3})\\ =&8\pi-\frac{8\pi}{1+4\lambda^2r^2}-8\pi\mathcal{J}(a)\frac{1}{\lambda^2}\\ &+\frac{1}{2r}\int_{\partial\mathbb{D}_{2r}}x^a(\nabla_{x^a}\pi\cdot j+\pi\cdot\nabla_{x^a}j+\nabla_aj\cdot j)+\mathcal{O}(\frac{1}{\lambda^3}). \end{align}\] Now on \(\Sigma\setminus U_{2r}\) we calculate \[\begin{align} \int_{\Sigma\setminus U_{2r}}|\nabla z|^2 =&\frac{2\pi}{\lambda^2r^2}-\frac{1}{2r}\int_{\partial\mathbb{D}_{2r}}x^a(\nabla_{x^a}\pi\cdot j+\pi\cdot\nabla_{x^a}j+\nabla_aj\cdot j)+\mathcal{O}(\frac{1}{\lambda^3}). \end{align}\] Summing these two equations completes the proof.
School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
A.M.Roberts@leeds.ac.uk