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\title{On the existence of degenerate solutions of the two-dimensional $H$-system}
\alttitle{Sur l'existence de solutions dÃ©gÃ©nÃ©rÃ©es du $H$-systÃ¨me en dimension deux}

\author{\firstname{Andr\'e } \lastname{Guerra}\CDRorcid{0000-0002-7226-7236}}
\address{Institute for Theoretical Studies, ETH ZÃ¼rich, CLV, Clausiusstrasse 47, 8006 ZÃ¼rich, Switzerland}
\email{andre.guerra@eth-its.ethz.ch}

\author{\firstname{Xavier} \lastname{Lamy}\CDRorcid{0000-0002-5281-0430}}
\address{Institut de MathÃ©matiques de Toulouse, UniversitÃ© Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 8, France}
\email{xavier.lamy@math.univ-toulouse.fr}

\author{\firstname{Konstantinos} \lastname{Zemas}\CDRorcid{0000-0002-3737-372X}}
\address{Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany}
\email{zemas@iam.uni-bonn.de}

\subjclass{35J47, 53A10}

\keywords{\kwd{$H$-system} \kwd{harmonic maps} \kwd{conformal maps}}
\altkeywords{\kwd{$H$-systÃ¨me} \kwd{applications harmoniques} \kwd{applications conformes}}

\thanks{AG was supported by Dr.\ Max R\"ossler, the Walter H\"afner Foundation and ETH Z\"urich Foundation. XL was supported by the ANR project ANR-22-CE40-0006. KZ was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Sonderforschungsbereich (SFB) 1060 and the Hausdorff Center for Mathematics (HCM) under Germany's Excellence Strategy EXC-2047/1-390685813}
\CDRGrant[ANR]{ANR-22-CE40-0006}
\CDRGrant[Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)]{1060}
\CDRGrant[Hausdorff Center for Mathematics (HCM)]{EXC-2047/1-390685813}

\begin{abstract}
We consider entire solutions $\omega\in\dot H^1(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system
\begin{equation*}
\Delta\omega=2\omega_x\wedge\omega_y\,
\end{equation*}
which we refer to as \emph{bubbles}. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.
\end{abstract}

\begin{altabstract}
Nous considÃ©rons des solutions entiÃ¨res $\omega\in\dot H^1(\mathbb{R}^2;\mathbb{R}^3)$ du $H$-systÃ¨me
\begin{equation*}
\Delta\omega=2\omega_x\wedge\omega_y,
\end{equation*}
appelÃ©es \emph{bulles}. De maniÃ¨re surprenante, et contrairement Ã&nbsp; des conjectures exprimÃ©es dans la littÃ©rature, nous montrons que les bulles de degrÃ© au moins trois peuvent Ãªtre dÃ©gÃ©nÃ©rÃ©es: le $H$-systÃ¨me linÃ©arisÃ© autour d'une bulle peut admettre des solutions qui ne soient pas tangentes Ã&nbsp; la famille lisse des bulles. Nous donnons de plus une caractÃ©risation algÃ©brique complÃ¨te des bulles dÃ©gÃ©nÃ©rÃ©es.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

In this short note we study critical points of the functional $\E\colon \dot H^1(\R^2 ;\R^3)\to \R$ defined by
\begin{equation}\label{eq:H_functional}
\E(u)\coloneqq \frac{1}{2}\int_{\R^2}\abs{\n u}^2 \d \mc L^2+\frac 2 3\int_{\R^2} \langle u, u_x\wedge u_y \rangle \d \mc L^2 ,
\end{equation}
where $\langle\,\cdot\,,\cdot\,\rangle$ denotes the standard inner product in $\R^3$, $\mc L^2$ the 2-dimensional Lebesgue measure and $u_x,u_y$ the partial derivatives of $u$ with respect to $x,y$ respectively. Both terms in~\eqref{eq:H_functional} are \emph{conformally invariant} and so, after identifying $\overline{\R^2}\cong \mb S^2$ through stereographic projection, the term
\begin{equation}\label{eq:vol}
\mc V(u)\coloneqq \frac 1 3 \int_{\R^2} \langle u,u_x \wedge u_ y\rangle \d \mc L^2
\end{equation}
corresponds to the signed algebraic volume of the region enclosed by $u(\mb S^2)$, whenever $u$ is regular. For maps which are just in the homogeneous Sobolev space $\dot H^1(\R^2;\R^3)$, one can define $ \mc V(u)$ through its continuous extension~\cite{Wente1969}.

The functional~\eqref{eq:H_functional} is very classical, as it appears naturally in the study of constant mean curvature surfaces, see e.g.~\cite[Section~III.5]{Struwe2008} and the references therein. To see this connection, we note that the first variation $\E'\colon \dot{H}^{1}(\R^2;\R^3) \to \dot H^{\,-1}(\R^2;\R^3)$ is given by
\begin{equation}\label{eq:first_variation}
\E'(u)=-\Delta u+2u_x\wedge u_y,
\end{equation}
and thus we arrive at the following:

\begin{definition}\label{def:bubbles}
A \emph{bubble} is a map $\omega \in \dot H^1(\R^2;\R^3)$ such that $\E'(\omega)=0$, i.e.,
\begin{equation}\label{eq:equation_for_bubbles}
\Delta \omega=2\,\omega_x\wedge \omega_y \;\text{ in }\; \mathcal{D}'(\R^2).
\end{equation}
\end{definition}

In other words, bubbles are simply entire finite-energy solutions of~\eqref{eq:equation_for_bubbles}, which is known as the \emph{$H$-system}. Any such solution is necessarily \emph{weakly conformal}, i.e., it satisfies
\begin{equation}\label{eq:omega_conf}
\abs{\omega_x}^2-\abs{\omega_y}^2 =\langle \omega_x,\omega_y\rangle =0 \;\text{ in }\; \R^2.
\end{equation}
Combined, equations~\eqref{eq:equation_for_bubbles} and~\eqref{eq:omega_conf} assert that $\omega$ is a (possibly branched) weakly conformal parametrization of a closed surface with mean curvature identically 1. In fact, a classical theorem due to Hopf asserts that any such surface is a unit sphere. This can also be seen as a consequence of the following classification result, due to Brezis and Coron~\cite[Lemma~A.1]{Brezis1985}, which describes completely the collection of bubbles:

\begin{theorem}[Classification of bubbles~\cite{Brezis1985}]\label{thm:BC}
Let $\omega \in \dot H^1(\R^2;\R^3)$ be a bubble. Then there exist complex polynomials $P,Q\in \C[z]$ and a vector $b\in \R^3$ such that
\begin{equation}\label{form_of_omega}
\omega(z)=\pi\Biggl(\frac{P(z)}{Q(z)}\Biggr)+b,
\end{equation}
where $\pi\colon \C \to \S^2$ is the inverse stereographic projection, i.e.,
\begin{equation}\label{eq:stereo}
\pi(z)\coloneqq \Biggl(\frac{2z}{\abs{z}^2+1},\frac{\abs{z}^2-1}{\abs{z}^2+1}\Biggr),
\end{equation}
and where we identify $ z=(x,y)=x+i y$. Moreover, if
\begin{equation}\label{eq:condPQ}
P/Q \;\text{ is irreducible}, \quad k\coloneqq \max \bigl\{\deg P,\deg Q \bigr\},
\end{equation}
then we have
\[
\frac{1}{8\pi}\int_{\R^2} \abs{\n \omega}^2 \d \mc L^2 = k.
\]
\end{theorem}

We refer to $k\in \N$ as in Theorem~\ref{thm:BC} as the \emph{degree} of $\omega$, since it coincides with the topological degree of $\omega$, viewed as a map between two-dimensional unit spheres.

Theorem~\ref{thm:BC} shows that the collection of bubbles can be seen as a disjoint union of the smooth, finite-dimensional manifolds
\begin{equation}\label{eq:M_k}
\mc M_k\coloneqq \biggl\{(P,Q,b)\in \C[z]\times \C[z]\times \R^3: \text{$P,Q$ are non-zero, $P$ is monic and~\eqref{eq:condPQ} holds}\biggr\}.
\end{equation}
An important problem is to understand the behavior of $\E$ near its critical points. To be more concrete, for any bubble $\omega\in \mc M_k$, we regard the second variation of $\E$ at $\omega$ as a linear operator $\E''(\omega)\colon \dot H^1(\R^2;\R^3)\to \dot H^{\,-1}(\R^2 ;\R^3)$. In view of~\eqref{eq:first_variation} it is explicitly given by
\begin{equation}\label{eq:linearized_operator}
\E''(\omega)[u]\coloneqq \restr{\frac{\d}{\d t}}{t=0}\E'(\omega+tu)=-\Delta u+2\bigl(\omega_x\wedge u_y+u_x\wedge \omega_y\bigr).
\end{equation}
Variations tangent to $\mc M_k$ at $\omega$ generate elements in the kernel of $\E''(\omega)$, so that
\[
\dim \ker \E''(\omega)\geq \dim \mc M_k = 4k+5.
\]
This leads us to the following standard concept:

\begin{definition}
A bubble $\omega$ with degree $k$ is said to be \emph{degenerate} if\/ $\dim \ker \E''(\omega)&gt;4k+5,$ and it is said to be \emph{non-degenerate} otherwise.
\end{definition}

As each $\mc M_k$ is smooth, non-degeneracy is equivalent to the usual notion of integrability, see e.g.~\cite[Section~3.13]{Simon1996}. Due to the conformal invariance of $\E$, we can and will regard each bubble as a map $\omega\colon \S^2\to \R^3$.

In~\cite[Lemma~5.5]{Isobe2001a} it was shown that bubbles with degree one are non-degenerate, see further~\cite[Appendix]{Chanillo05} and~\cite[Section~3]{Musina04}, while in~\cite[Theorem~1.1]{SireWeiZhengZhou23} it was shown that the \emph{standard $k$-bubble}, corresponding to the choice $P(z)=z^k$ and $Q(z)=1$ in~\eqref{form_of_omega}, is non-degenerate as well. These works also raise the conjecture that all bubbles should be non-degenerate, cf.~\cite[p.\ 190]{Chanillo05} and~\cite[p.\ 4]{SireWeiZhengZhou23}. Surprisingly, in this note we observe that although this is generically the case, for $k\geq 3$ there is an \emph{exceptional set of degenerate bubbles} in $\mc M_k$:

\begin{theorem}[Characterization of degenerate bubbles]\label{thm:charact}
Let $\omega\colon \S^2\to \R^3$ be a bubble as in~\eqref{form_of_omega} whose set of branch points is given by
\[
\{\abs{\n \omega} = 0 \} \eqqcolon  \{p_1, \dots,p_n\}.
\]
Let $z$ be a conformal coordinate on $\S^2$ with respect to which none of the branch points is $\infty$. Then $\omega$ is degenerate if and only if there exists a non-zero polynomial $R\in \C[z]$ with $\deg R \leq n-4$ such that the meromorphic function $h\colon \C\to \C$ defined by
\begin{equation*}
h(z) \coloneqq  \frac{R(z)}{(z-p_1)\dots(z-p_n)}
\end{equation*}
solves the algebraic system of equations
\[
\Res_{p_j} \Biggl(\frac{h}{(P/Q)'}\Biggr) = 0 \quad \text{ for }\; j\in \{1,\dots, n\}.
\]
\end{theorem}

We note that Theorem~\ref{thm:charact} implies that the set of degenerate bubbles of degree $k$ has (real) codimension at least two in $\mc M_k$, see the comment after~\cite[Theorem~2]{EjiriKotani92}. The above characterization follows from a correspondence between nontrivial elements in the kernel of $\E''[\omega]$ and solutions of a well-studied SchrÃ¶dinger equation on $\S^2$, cf.~\cite{Ejiri94,Ejiri98,EjiriKotani92,EjiriKotani93,Kotani97,MontielRos91,Nayatani93}. As immediate consequences of Theorem~\ref{thm:charact} we obtain the following:

\begin{corollary}[Examples of non-degenerate bubbles]\label{cor:exnondeg}
Let $\omega$ be a degree $k$ bubble as in~\eqref{form_of_omega}. If $\omega$ is degenerate then it has at least 4 branch points. In particular:
\begin{enumerate}
\item\label{it:deg2}
if\/ $k\leq 2$ then $\omega$ is non-degenerate;
\item\label{it:standard}
the standard $k$-bubble corresponding to
\[
P(z)=z^k, \quad Q(z)=1,
\]
is non-degenerate for all $k\in \N$.
\end{enumerate}
\end{corollary}

Other non-degenerate examples can be inferred from~\cite{MontielRos91}, see e.g.~\cite[Corollary~15]{MontielRos91}.

\begin{corollary}[Examples of degenerate bubbles]\label{cor:exdeg}
Let $\omega$ be a degree $k$ bubble as in~\eqref{form_of_omega}. If\/ $k=3$ then $\omega$ is degenerate if and only if
\[
P(z) = z^3+2, \quad Q(z) = z,
\]
up to a M\"obius transformation of the sphere.
\end{corollary}

An interesting question that can subsequently be pursued is to prove optimal \L ojasiewicz inequalities for degenerate bubbles. There are only a few examples of variational problems where this is achieved~\cite{Frank2023,Frank2023a}. We also refer the reader to~\cite{Malchiodi2020} for a related result, where \L ojasiewicz inequalities near the simplest possible bubble tree in a surface with positive genus were obtained. Such inequalities determine in a precise way the leading order behavior of $\E$ near a bubble, and this information is useful in arguments involving blow-up or perturbative analysis, see e.g.~\cite{Brezis1985,Caldiroli04,Caldiroli04reduc,Caldiroli2006,Chanillo05,Isobe2001a, Isobe2001b,SireWeiZhengZhou23,Struwe1985}.

\section{Characterization of degenerate bubbles}

As explained in the introduction, the non-degeneracy of bubbles (cf.~Definition~\ref{def:bubbles}) is characterized through the study of the kernel of the linearized operator $\E''(\omega)\colon\dot H^{1}(\R^2;\R^3)\to \dot H^{\,-1}(\R^2;\R^3)$ defined in~\eqref{eq:linearized_operator}. In other words, we are interested in classifying the space of solutions $u\in \dot H^1(\R^2;\R^3)$ to
\begin{align}\label{eq:linHsyst}
\Delta u =2(\omega_x\wedge u_y +u_x\wedge \omega_y)\;\text{ in }\; \mathcal{D}'(\R^2)\,
\end{align}
for an arbitrary bubble $\omega$. We note that, by elliptic regularity, any such solution is necessarily smooth.

Given a bubble $\omega$ as in~\eqref{form_of_omega}, we can assume without loss of generality that $b=0$, since $\E$ is invariant under translations in the target. Then, due to the conformal invariance of $\E$, we can regard bubbles as maps $\omega \colon \mb S^2\to \mb S^2\subset \R^3$. More explicitly, given a bubble $\omega$, we will write
\begin{equation}\label{eq:defomega}
\omega(z) = \pi\bigl(\varphi(z)\bigr), \qquad \varphi(z) \coloneqq  \frac{P(z)}{Q(z)}, \qquad \deg P \leq \deg Q,
\end{equation}
where $\pi$ is as in~\eqref{eq:stereo} and $P,Q\in \C[z]$ for some conformal coordinate $z$ on $\mb S^2$; the assumption on the degrees of $P,Q$ can always be achieved by choosing a suitable coordinate $z$.

We find it convenient to use complex notation, and henceforth we will write
\[
\p_z \coloneqq  (\p_x - i \p_y)/2, \quad \p_{\bar z} \coloneqq  (\p_x + i \p_y)/2
\]
for the usual Wirtinger derivatives. Let the Euclidean inner product $\langle\,\cdot\,, \cdot\,\rangle$ in $\R^3$ be extended as a complex-bilinear form on $\C^3$. Then the fact~\eqref{eq:omega_conf} that each bubble is weakly conformal is expressed concisely as
\begin{equation}\label{eq:holomw}
\langle \omega_z,\omega_z\rangle = 0 \;\text{ in }\; \C,
\end{equation}
and the linearization of this equation leads us, as in~\cite[Section~2]{Gulliver89}, to the following:

\begin{definition}
Given a bubble $\omega\colon\S^2\to \S^2$ and a vector field $v\in \dot H^1(\S^2;\R^3)$, we say that $v$ is a \emph{conformal Jacobi field along} $\omega$ if $\langle \omega_z,v_z\rangle = 0$.
\end{definition}

We can also express equations~\eqref{eq:equation_for_bubbles} and~\eqref{eq:linHsyst} respectively as
\begin{align}
\omega_{z\bar z} &amp; = i\, \omega_{\bar z}\wedge \omega_z,\label{eq:complexHsys}\\
u_{z \bar z} &amp; = i (u_{\bar z}\wedge \omega_z + \omega_{\bar z} \wedge u_z),\label{eq:complexlinHsys}
\end{align}
in complex notation. We then have the following:

\begin{lemma}\label{lem:confjacobi}
Let $u\in \dot H^1(\S^2; \R^3)$ solve~\eqref{eq:complexlinHsys} for a bubble $\omega$. Then $u$ is a conformal Jacobi field.
\end{lemma}

\begin{proof}
We use~\eqref{eq:complexHsys} and~\eqref{eq:complexlinHsys} to compute
\begin{equation*}
\begin{split}
\p_{\bar z} \langle \omega_z,u_z\rangle &amp; = i \bigl(\langle \omega_{\bar z} \wedge \omega_z, u_z\rangle + \langle \omega_z, u_{\bar z} \wedge \omega_z+ \omega_{\bar z}\wedge u_z \rangle\bigr) \\
&amp; = i \bigl(\langle \omega_{\bar z} \wedge \omega_z, u_z\rangle + \langle \omega_z, \omega_{\bar z}\wedge u_z \rangle \bigr)\\
&amp;= 0,
\end{split}
\end{equation*}
where the last equality is simply an algebraic identity resulting from $\langle a, b\wedge c\rangle = \det(a\vert b\vert c)$, where $(a\vert b\vert c)$ denotes the $3\times 3$ matrix with column vectors $a,b,c$ (in this order). Thus $\langle \omega_z, u_z\rangle$ is integrable and holomorphic, hence by Liouville's theorem it is zero.
\end{proof}

Note that each bubble $\omega$ as in~\eqref{eq:defomega} is a \emph{harmonic map} into $\mb S^2$, i.e.,
\begin{equation}\label{eq:harmonic_map}
\Delta \omega+\abs{\nabla \omega}^2\omega=0 \quad \iff \quad \omega_{z \bar z} +\abs{\omega_z}^2 \omega =0,
\end{equation}
as can be checked directly by differentiating $\abs{\omega}=1$, using the weak conformality~\eqref{eq:omega_conf} of $\omega$. Since the target is a 2-sphere, $\omega$ is an integrable harmonic map~\cite{Gulliver89}; we note that this integrability does not hold for higher-dimensional spheres~\cite{Lemaire2009}. The following proposition is essentially a consequence of this integrability result.

\begin{proposition}[Decomposition]\label{prop:decomp}
Let $\omega\colon \S^2\to \S^2$ be a degree $k$ bubble and decompose maps $u\in\ker\E''(\omega)$ according to
\begin{equation*}
\ker \E''(\omega) = T(\omega)\oplus N(\omega)\,\omega,
\qquad u = \bigl[u - \langle u,\omega \rangle \omega\bigr]+ \langle u,\omega \rangle \omega,
\end{equation*}
where this decomposition is orthogonal pointwise with respect to the inner product of\/ $\R^3$. The elements of\/ $T(\omega)$ are generated by infinitesimal variations in the coefficients of $\varphi$ in~\eqref{eq:defomega}, so that $\dim T(\omega) = 4k+2$, and
\begin{equation}\label{eq:defN}
N(\omega) \coloneqq  \bigl\{f\in C^\infty(\S^2;\R): \Delta f + \abs{\nabla \omega}^2 f=0\bigr\}.
\end{equation}
\end{proposition}

\begin{proof}
Let us note that the equation defining $N(\omega)$ can be written in complex notation as
\begin{equation}\label{eq:complexSchrod}
f_{z\bar z } + \abs{\omega_z}^2 f=0.
\end{equation}
We now observe that $f\in N(\omega)$ if and only if the map $f\omega$ solves~\eqref{eq:complexlinHsys}, since
\begin{equation*}
\begin{split}
(f \omega)_{z\bar z} - i \bigl((f \omega)_{\bar z} \wedge \omega_z + \omega_{\bar z} \wedge (f \omega)_z\bigr) &amp; = f_{z \bar z}\, \omega - i f \omega_{\bar z} \wedge \omega_z + f_z \omega_{\bar z} + f_{\bar z} \omega_z - i \bigl(f_{\bar z} \omega \wedge \omega_z + f_z \omega_{\bar z} \wedge \omega\bigr)\\
&amp; = \bigl[f_{z \bar z} +\abs{\omega_z}^2 f \bigr]\omega,
\end{split}
\end{equation*}
where we used~\eqref{eq:complexHsys} and the identities
\begin{equation}\label{eq:omega_identities}
\omega_z = i \,\omega\wedge \omega_z, \ \omega_{\bar z} = i\, \omega_{\bar z} \wedge \omega, \;\text{ and }\; i \,\omega_z \wedge \omega_{\bar z} = \abs{\omega_z}^2 \omega,
\end{equation}
which follow from the fact that $\biggl(\frac{\sqrt 2\omega_x}{\abs{\nabla \omega}},\frac{\sqrt 2\omega_y}{\abs{\nabla \omega}}, -\omega\biggr)$ is a positively oriented orthonormal frame of $\R^3$. In view of~\eqref{eq:complexlinHsys}, $N(\omega)\omega\subset \ker \E''(\omega)$, and reversely, for any $u \in \ker \E''(\omega)$ using~\eqref{eq:complexHsys}, \eqref{eq:complexlinHsys} and~\eqref{eq:omega_identities} we have $\langle u,\omega\rangle \in N(\omega)$, and so from the above, $\langle u,\omega\rangle\omega \in \ker \E''(\omega)$. Note that, by elliptic regularity, any $f\in \dot H^1(\S^2; \R)$ solving~\eqref{eq:complexSchrod} is actually smooth.

It remains to prove the characterization of $T(\omega)$. For this, we first note that since $\abs{\omega}^2=1$ we have $\langle \omega_z,\omega\rangle = 0$. Thus, by~\eqref{eq:holomw}, for any $f\in \dot H^1(\S^2;\R)$ the vector field $f\omega$ is a conformal Jacobi field along $\omega$. By Lemma~\ref{lem:confjacobi}, any $u\in \ker \E''(\omega)$ is also a conformal Jacobi field along $\omega$, and hence by linearity the same also holds for the map $\tilde u \coloneqq  u - \langle u, \omega\rangle\omega$. Moreover, note that $\tilde u(z) \in T_{\omega(z)}\S^2$. Let us assume that the polynomials $P,Q$ in~\eqref{eq:defomega} satisfy $\det P \leq \deg Q$, the other case being entirely analogous. Applying~\cite[Lemma~3]{Gulliver89}, we deduce that $\tilde u$ is integrable, i.e., there are polynomials $A,B\in \C[z]$ such that
\begin{equation}\label{eq:tildeu}
\tilde u = \restr{\frac{\d}{\d t}}{t=0} \pi\Biggl(\frac{P+t A}{Q+tB}\Biggr),
\end{equation}
where $\deg (A)\leq \deg(P)$ and $\deg(B)&lt;\det(Q)$. Conversely, given polynomials $A,B$ satisfying these conditions, we can define a map $\tilde u\in T(\omega)$ through~\eqref{eq:tildeu}. This completes the characterization and thus $\dim T(\omega)=4 k+2$.
\end{proof}

\begin{corollary}\label{cor:nullity}
We have $\dim N(\omega)\geq 3$ with equality if and only if $\omega$ is non-degenerate.
\end{corollary}

\begin{proof}
By~\eqref{eq:harmonic_map}, the linear space
\begin{equation}\label{eq:defL}
L(\omega) \coloneqq  \bigl\{ \langle \xi, \omega \rangle :\xi \in \R^3\bigr\}
\end{equation}
is a subspace of $N(\omega)$, and thus $\dim N(\omega)\geq 3$. By Proposition~\ref{prop:decomp}, equality holds if and only if $\dim \ker \E''(\omega)=4k+5$, i.e., if and only if $\omega$ is non-degenerate.
\end{proof}

We refer to $L(\omega)$ as the space of \emph{trivial} solutions to the Schr\"odinger equation~\eqref{eq:complexSchrod}. Since $\omega$ is conformal, \eqref{eq:complexSchrod} can be rewritten as
\begin{equation*}\label{eq:eigenvalue_eqn_sing_metric}
\Delta_{g_\omega}\tilde f+2 \tilde f=0, \;\text{ where }\; g_\omega\coloneqq \omega^* g_0 \;\text{ and }\; \tilde f \coloneqq  f \circ \pi^{-1}\in \dot{H}^1(\S^2 ;\R).
\end{equation*}
Here $g_\omega$ is the pullback metric of the standard round metric $g_0$ on $\mathbb{S}^2$ induced by $\omega\colon \S^2\to \S^2$; the metric $g_\omega$ in general has \emph{conical singularities} at the branch points of $\omega$. Equation~\eqref{eq:complexSchrod} has received a lot of attention in the last decades, and the space of solutions is completely understood, see e.g.~\cite{Ejiri94,Ejiri98,EjiriKotani92,EjiriKotani93,Kotani97,MontielRos91,Nayatani93}. Surprisingly, for non-generic $\omega$ there are non-trivial solutions to~\eqref{eq:complexSchrod}, i.e.\ $\dim N(\omega)&gt;3$; the space of such $\omega$ has codimension larger than one~\cite[Proof of Theorem~A]{EjiriKotani93}. In general, $\dim N(\omega)-3$ is \emph{even}, since $N(\omega)/L(\omega)$ is a complex vector space~\cite[Proposition~18]{MontielRos91}, and so we can write $\dim N(\omega)=3+2d$ for $d\in \N_0$. The $2d$ extra directions can be interpreted as arising from smooth one-parameter families of harmonic maps with values in higher dimensional spheres~\cite[Theorem~A]{Ejiri98}.

The following result provides a concrete algebraic characterization of those bubbles $\omega$ for which there are nontrivial elements in $N(\omega)$.

\begin{theorem}[\cite{EjiriKotani93,MontielRos91}]\label{thm:MR}
Let $\omega$ be a bubble as in~\eqref{eq:defomega}, let $\{p_1, \dots, p_n\}\subset \S^2$ be its set of branch points and $z$ a conformal coordinate on $\mathbb{S}^2$ with respect to which none of the branch points is $\infty$. There is a linear bijection between the space $N(\omega)/ L(\omega)$ of non-trivial solutions to~\eqref{eq:complexSchrod} and the space of non-zero polynomials $R\in \C[z]$ with $\deg R \leq n-4$ and such that
\begin{equation}\label{eq:condsR}
\Res_{p_j} (h/\varphi')= 0 \;\text{ for all }\; j\in \{1, \dots, n\}, \;\text{ where }\; h(z)\coloneqq  \frac{R(z)}{(z-p_1)\cdots(z-p_n)}.
\end{equation}
\end{theorem}

The proof of Theorem~\ref{thm:MR} in the above references is carried out in a more general context. In order to keep the exposition mostly self-contained, we include here a direct proof of their result in our setting, following essentially the arguments in~\cite{MontielRos91}.

The main idea behind Theorem~\ref{thm:MR} is to use the \emph{Gauss parametrization}~\cite{Dajczer1985}. A surface $S\subset \R^3$ with invertible Gauss map $\omega$ is locally parametrized by an immersion
\begin{equation}\label{eq:defX}
X=f\omega + \nabla_{g_\omega} f = f\omega +\frac{1}{\abs{\omega_z}^2}\bigl(f_z\omega_{\bar z}+f_{\bar z}\omega_z\bigr),
\end{equation}
where $f=\langle X, \omega\rangle$ is the \emph{support function} of $S$. Thus, in the setting of Theorem~\ref{thm:MR}, we have a local correspondence between functions $f\colon \mb S^2\setminus \{p_1, \dots, p_n\}\to \R$ and punctured surfaces with generalized Gauss map $\omega$. The next lemma gives a geometric interpretation of the condition $f\in N(\omega)$ (recall~\eqref{eq:defN}) in terms of the corresponding immersion defined through~\eqref{eq:defX}.

\begin{lemma}\label{lemma:gaussparam}
Let $\omega$ be a bubble with branch points $\{p_1, \dots, p_n\}\subset \S^2$, let $f\in C^\infty(\S^2;\R)$ and let $X\colon \S^2\setminus \{p_1, \dots, p_n\}\to \R^3$ be defined as in~\eqref{eq:defX}. Then
\[
f\in N(\omega) \quad \iff \quad \text{$X$ is minimal, i.e., it is weakly conformal and harmonic}.
\]
\end{lemma}

\begin{proof}
Using~\eqref{eq:holomw} and $\abs{\omega}=1$, after differentiation we deduce that
\begin{equation}\label{eq:iii}
\omega_{zz} =\bigl[\log\bigl(\abs{\omega_z}^2\bigr)\bigr]_z \,\omega_z.
\end{equation}
An elementary but lengthy calculation, using~\eqref{eq:iii}, then shows that
\begin{equation}\label{eq:iv}
\bigl[\log\bigl(\abs{\omega_z}^2\bigr)\bigr]_{z\bar z} + \abs{\omega_z}^2=0.
\end{equation}
Note also that $\p_z \biggl(\frac{1}{\abs{\omega_z}^2}\biggr) = - \frac{\log(\abs{\omega_z}^2)_z}{\abs{\omega_z}^2}$; this, combined with~\eqref{eq:harmonic_map} and~\eqref{eq:iv}, yields
\begin{equation}\label{eq:X_z}
X_z = \bigl(f_{z \bar z} + \abs{\omega_z}^2 f\bigr)\frac{\omega_z}{\abs{\omega_z}^2} + h_f \frac{\omega_{\bar z}}{\abs{\omega_z}^2},
\end{equation}
where we set
\begin{align}\label{eq:defh}
h_f\coloneqq f_{zz}-\bigl[\log\bigl(\abs{\omega_z}^2\bigr)\bigr]_z f_z.
\end{align}
We can then further compute
\begin{equation}\label{eq:confX}
\langle X_z, X_z \rangle = 2\frac{h_f}{\abs{\omega_z}^2} \bigl(f_{z \bar z} + \abs{\omega_z}^2 f\bigr).
\end{equation}

The lemma follows from the above identities using elementary calculations. If $f\in N(\omega)$ then clearly $X$ is weakly conformal by~\eqref{eq:confX} and one can also check that $X_{z \bar z}=0$, thus $X$ is minimal. Conversely, if $X$ is minimal then $f\in N(\omega)$: this can be deduced from~\eqref{eq:X_z} since
\[
\langle X_{z \bar z}, \omega\rangle = -\bigl(f_{z \bar z} + \abs{\omega_z}^2 f\bigr),
\]
so if $X$ is harmonic then $f\in N(\omega)$.
\end{proof}

It will be useful to convert between $\omega$ and $\varphi$ through the stereographic projection, as in~\eqref{eq:defomega}. Using the fact that $\varphi$ is meromorphic, one can verify that
\begin{equation}\label{eq:derivatives}
\frac{\omega_{\bar z}}{\abs{\omega_z}^2} = \frac{1}{\varphi'}\Biggl(\frac{1-\varphi^2}{2},\frac{i(1+\varphi^2)}{2},\varphi\Biggr).
\end{equation}
The next lemma is a simple consequence of the above calculations.

\begin{lemma}\label{lemma:h}
For any conformal coordinate $z$ on $ \mathbb S^2$ and $f\in N(\omega)$, let $h_f\colon \mb S^2\setminus \{p_1,\dots,p_n\}\to \R$ be defined as in~\eqref{eq:defh}. Then:
\begin{enumerate}
\item \label{it:hismero}
$h_f$ is meromorphic, i.e., $\p_{\bar z} \,h_f=0$ on $\mathbb S^2\setminus \lbrace p_1,\dots,p_n\rbrace$, and its poles are simple;
\item\label{it:zeroresidue}
the meromorphic function $h_f/\varphi'$, where $\varphi=P/Q$, has zero residue at each pole $p_j$.
\end{enumerate}
\end{lemma}
\begin{proof}
For~\eqref{it:hismero} we note that, since $f\in N(\omega)$, from~\eqref{eq:X_z} and~\eqref{eq:derivatives} we obtain
\begin{equation}\label{eq:X_z-varphi}
X_z =h_f \frac{\omega_{\bar z}}{\abs{\omega_z}^2}= \frac{h_f}{\varphi'} \Biggl(\frac{1-\varphi^2}{2},\frac{i(1+\varphi^2)}{2},\varphi\Biggr).
\end{equation}
Thus $\p_{\bar z} h_f =0$, since $\varphi$ is meromorphic and $X_{z \bar z}=0$ according to Lemma~\ref{lemma:gaussparam}. Moreover, the poles of $h_f$ are simple, since in~\eqref{eq:defh} $f$ is smooth and $\abs{\omega_z}$ vanishes to finite order at each branch point. Claim~\eqref{it:zeroresidue} follows again from~\eqref{eq:X_z-varphi}, which implies that $h_f/\varphi'\d z= (X^1-i X^2)_z \d z$: the latter differential has zero integral along any loop, and hence zero residue at each pole.
\end{proof}

\begin{corollary}\label{cor:h}
For $f\in N(\omega)$ and any choice of conformal coordinate $z$ with respect to which none of the branch points is $\infty$, there exists $R\in \mathbb C[z]$ of degree at most $n-4$ such that
\begin{equation}\label{eq:Rhf}
h_f(z)=\frac{R(z)}{(z-p_1)\cdots (z-p_n)}.
\end{equation}
\end{corollary}
\begin{proof}
Applying a translation in the complex plane if necessary, we may without restriction fix a conformal coordinate $w$ in which none of the poles of $\varphi$ is $0$ or $\infty$. By Lemma~\ref{lemma:h}\eqref{it:hismero}, there exists an entire function $g$ and $a_1,\dots,a_n\in\C$ such that
\begin{equation*}
h_f(w)=g(w) +\sum_{j=1}^n \frac{a_j}{w-p_j}.
\end{equation*}
If $w=w(z)$ is a conformal change of variables then the meromorphic quadratic differential $h_f(w)(\d w)^2$ pulls back to $h_f\bigl(w(z)\bigr) (\d w/\d z)^2 (\d z)^2.$ Thus, when $w(z)=1/z$, we obtain a new meromorphic function
\begin{equation*}
h_f\bigl(w(z)\bigr) \Biggl(\frac{\d w}{\d z}\Biggr)^2 = \frac{1}{z^4}g(1/z) +\frac{1}{z^4}\sum_{j=1}^n \frac{a_j}{1/z -p_j} =\frac{1}{z^4}g(1/z) +\frac{1}{z^3}\frac{\tilde R(z)}{\prod_{j=1}^{ n}(z-1/p_j)},
\end{equation*}
for some $\tilde R\in\mathbb C[z]$ with $\deg \tilde R\leq n-1$. In these new coordinates, Lemma~\ref{lemma:h}\eqref{it:hismero} implies that the entire function $g$ is zero and that $\tilde R(z)=z^3 R(z)$ for some $R\in\mathbb C[z]$ with $\deg R\leq n-4$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:MR}]
By Lemma~\ref{lemma:h} and Corollary~\ref{cor:h}, for each $f\in N(\omega)$ one can associate a function $h_f\colon \mb S^2\setminus \{p_1,\dots,p_n\}\to \R$ satisfying~\eqref{eq:condsR}. It is easy to see from~\eqref{eq:iii} and~\eqref{eq:defh} that if $f\in L(\omega)$ then $h_f=0$, i.e., the corresponding polynomial $R$ in~\eqref{eq:Rhf} is zero.

Conversely, given a function $h\colon \mb S^2\setminus \{p_1,\dots,p_n\}\to \R$ satisfying~\eqref{eq:condsR}, one constructs an immersion $X_h\colon \mb S^2\setminus \{p_1,\dots,p_n\}\to \R^3$ such that
\[
\p_z X_h = h \frac{\omega_{\bar z}}{\abs{\omega_z}^2},
\]
cf.~\eqref{eq:X_z-varphi}. In fact, we construct $X_h$ by
\begin{equation}\label{eq:intX}
X_h = \Real \int \p_z X_h \d z,
\end{equation}
where the path starts from a fixed point on $\mb S^2\setminus \{p_1,\dots,p_n\}$; the no residue condition in~\eqref{eq:condsR} guarantees that $X_h$ is independent of the choice of path. Moreover, $X_h$ is unique up to addition of a constant vector in $\R^3$. We claim that $\langle X_h, \omega\rangle \in N(\omega)$. Once this is shown the theorem follows, since the linear maps
\[
N(\omega)\ni f\mapsto f\omega + \frac{1}{\abs{\omega_z}^2} (f_z \omega_{\bar z} + f_{\bar z} \omega_z), \qquad X\mapsto \langle X, \omega\rangle\in N(\omega)
\]
are inverse to each other, and hence
\[
N(\omega)/L(\omega)\ni f\mapsto h_f, \qquad h\mapsto \langle X_h,\omega\rangle\in N(\omega)/L(\omega)
\]
are also inverses, since adding a constant to $X_h$ amounts to adding a trivial solution to $f$.

Thus, to complete the proof, it remains to show that $\langle X_h, \omega\rangle \in N(\omega)$. To be precise, note that $\langle X_h, \omega\rangle$ is only defined in $\mb S^2\setminus \{p_1,\dots, p_n\}$ and that, in this punctured sphere, it is a solution of the Schr\"odinger equation~\eqref{eq:complexSchrod}, since
\begin{equation*}
\begin{split}
\p_{z\bar z} \langle X_h, \omega\rangle + \abs{\omega_z}^2 \langle X_h, \omega\rangle &amp; = \p_{\bar z} \bigl(\langle \p_z X_h, \omega\rangle + \langle X_h,\omega_z\rangle \bigr) + \abs{\omega_z}^2 \langle X_h, \omega\rangle\\
&amp; =\langle X_h, \omega_{z\bar z}\rangle + \langle X_h, \abs{\omega_z}^2 \omega\rangle\\
&amp;=0.
\end{split}
\end{equation*}
The last equality follows from~\eqref{eq:harmonic_map} and to pass to the second line we used that
\[
\langle \partial_zX_h, \omega \rangle =\frac{h}{\abs{\omega_z}^2}\langle \omega_{\bar z}, \omega\rangle=0,
\]
since $\abs{\omega}=1$, and that, since $X_h$ is a real vector field,
\[
\langle \p_{\bar z} X_h, \omega_z\rangle = \langle \p_{\bar z} \bar{X_h}, \omega_z\rangle = \langle \bar{\p_z X_h}, \omega_z\rangle=\frac{h}{\abs{\omega_z}^2}\langle \omega_z, \omega_z\rangle=0,
\]
cf.~\eqref{eq:holomw}. Thus from elliptic regularity theory we deduce that $\langle X_h, \omega\rangle$ can be extended to a function in $N(\omega)$ once we show that $\langle X_h,\omega\rangle \in L^\infty\bigl(\mb S^2\setminus \{p_1, \dots,p_n\}\bigr).$

To prove the boundedness of $\langle X_h,\omega\rangle$, fix $p_j$ for some $j=1,\dots,n,$ and choose a local conformal coordinate $z$ in a neighborhood of $p_j$ with $z(p_j)=0$. Up to a rotation in $\R^3$, we can suppose that $\varphi(z) = z^{m_j}$ in this neighborhood, where $m_j&gt;1$, and so, from~\eqref{eq:derivatives}, we see that
\begin{equation}\label{def:X_z}
\p_z X_h = \frac{h}{m_j} \Biggl(\frac{z^{1-m_j} - z^{1+m_j}}{2}, \frac{i (z^{1-m_j} + z^{1+m_j})}{2}, z
\Biggr).
\end{equation}
In these coordinates we may write $h$ as $h(z)=\frac{c_0}{z}+\sum_{\ell=0}^\infty c_\ell z^\ell$, for some $(c_\ell)_{\ell\in \N}\in \C$, so that~\eqref{eq:intX} and~\eqref{def:X_z} imply that in a neighborhood of 0,
\[
X_h(z)=\frac{1}{2m_j(1-m_j)}\Bigl(c_0 \Real(z^{1-m_j})+o(z^{1-m_j}),c_0\Imag(z^{1-m_j})+o(z^{1-m_j}),c_0(1-m_j)2\Real(z)+o(z)\Bigr),
\]
where $\lim_{z\to 0}\frac{o(z)}{z}=0$. By~\eqref{eq:defomega} and~\eqref{eq:stereo} the bubble $\omega$ can be expressed in these coordinates as
\[
\omega(z)=\Biggl(\frac{2z^{m_j}}{\abs{z}^{2m_j}+1}, \frac{\abs{z}^{2m_j}-1}{\abs{z}^{2m_j}+1}\Biggr),
\]
hence the fact that $\langle X_h, \omega\rangle$ is bounded near 0 follows from the last two formulas.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:charact}]
The result follows by combining Corollary~\ref{cor:nullity} with Theorem~\ref{thm:MR}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:exnondeg}]
The main claim follows from Theorem~\ref{thm:charact}, since if $\omega$ is degenerate then there must exist a corresponding non-zero polynomial $R$ with $\deg R\leq n-4$, where $\omega$ has $n$ branch points, say $p_1, \dots, p_n$. Hence if $\omega$ is degenerate we must have $n\geq 4$. This also immediately implies~\ref{it:standard}. To prove~\ref{it:deg2} note that if $m_j&gt;1$ is the multiplicity of $\omega$ at $p_j$ (i.e.\ if $m_j-1$ is the algebraic multiplicity of $p_j$ as a zero of $\abs{\nabla \omega}$), then by the Riemann--Hurwitz formula we have
\begin{equation}\label{eq:RH}
2(k-1) = \sum_{j=1}^n (m_j -1),
\end{equation}
where $k$ is the degree of $\omega$. Thus $\omega$ can have at least 4 branch points only if $k\geq 3 $.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:exdeg}]
Note that, by~\eqref{eq:RH}, a degree~3 bubble can only have at most 4 different branch points, so by Corollary~\ref{cor:exnondeg} if $\omega$ is degenerate then indeed it must have exactly 4 different branch points $p_1,\dots, p_4$. By Theorem~\ref{thm:charact}, we see that $\omega$ is degenerate if and only if
\[
\Res_{p_j} (1/\varphi') = 0 \;\text{ for }\; j=1, 2,3,4,
\]
where $\varphi = P/Q$ is as in~\eqref{eq:defomega}. Elementary computations then yield the conclusion, see~\cite[p.\ 171]{MontielRos91} for further details.
\end{proof}

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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