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\begin{document}
\title{Prescribing mean curvature: existence and uniqueness problems}

\author{G. Kamberov} 
\address{Department of Mathematics,
Washington University, St. Louis, MO}
\email{kamberov@@math.wustl.edu}
\issueinfo{4}{02}{}{1998}
\dateposted{March 18, 1998}
\pagespan{4}{11}
\PII{S 1079-6762(98)00040-7}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}
\date{February 27, 1996}
\revdate{January 1, 1998}
\subjclass{Primary 53C42; Secondary 35Q40, 53A05, 53A30, 53A50, 58D10, 81Q05}
\keywords{Surfaces, spinors, conformal immersions, prescribing mean curvature}

%\maketitle

%{\footnotesize
\begin{abstract}
The paper presents results on the extent to which mean curvature data can 
be used to determine a surface in $\bR^{3}$ or its shape. 
The emphasis is on Bonnet's problem: classify and study the
surface immersions in $\bR^3$ whose shape is not uniquely determined by the
first fundamental form and the mean curvature function.  These immersions are
called Bonnet immersions.  A local solution of Bonnet's problem for
umbilic-free immersions follows from papers by Bonnet, Cartan,
and Chern. The properties of immersions with umbilics and global rigidity
results for closed surfaces are presented in the first part of this paper.

The second part of the paper outlines an existence theory for conformal
immersions based on Dirac spinors along with its immediate applications to
Bonnet's problem.  The presented existence paradigm provides insight into the
topology of the moduli space of Bonnet immersions of a closed surface, and  
reveals a  parallel between Bonnet's problem and Pauli's exclusion principle. 
\end{abstract}
%}

\maketitle

\section*{Introduction} 
The results in this paper are part of a program to understand 
to what extent one can use mean curvature data to 
determine the shape of a surface in $\bR^3$. 

The first section of the paper presents results on a
problem suggested by Bonnet. A generic immersion is
uniquely determined up to a rigid motion by its first fundamental form
and its mean curvature function, but there are some exceptions, for
example most constant mean curvature immersions.  Bonnet's problem is
to classify and study all exceptional immersions.  Here I concentrate on
the study of Bonnet's problem for immersions with umbilic points and
immersions of closed surfaces.  

The second section contains an outline of an existence theory for conformal
immersions (\cite{GANG,franz,ulrich,saopaulo}) along with its 
immediate applications to Bon\-net's problem.  The central idea in
the theory is to define square roots of basic geometric
objects, for example, area elements, differentials of maps, and then
determine the equations satisfied by the square root of the
differential of a conformal immersion.  From an analytic point of view
the main advantage of the approach is that the existence problem
for prescribing mean curvature data is reduced to a
first order elliptic system. The theory suggests a new paradigm: the
data used to determine the immersion are the conformal structure, the
regular homotopy class of the immersion, and the mean curvature
half-density, that is, the mean curvature appropriately weighted by the
metric.  These data determine the immersion via a generalized
Weierstrass-Kenmotsu formula.

Robert Bryant introduced me to Bonnet's problem. I thank him and Ulrich
Pinkall, Franz Pedit, Peter Norman, and Dennis DeTurck for many discussions, advice, and support. 

\section{Bonnet's uniqueness problem}
Two isometric immersions $F_1$ and $F_2$, of a given Riemann
surface $(M, g)$ into $\bR^3$, are called congruent (denoted by 
$F_1 \sim F_2$) if they differ by a rigid motion. I am 
interested in the following questions: What conditions guarantee that
$H_1 = H_2$ implies $F_1 \sim F_2$? What can be said if $H_1 = H_2$,
but $F_1$ and $F_2$ are not congruent? The isometric immersions $F_1$ and 
$F_2$ are called Bonnet mates if they are not congruent,
but their mean curvature functions, $H_1$ and $H_2$, are equal. 

Bonnet, Cartan, and Chern studied the existence and classification 
of Bonnet mates. Their works yield a complete local classification 
of the umbilic-free immer\-sions which admit Bonnet mates 
(\cite{Bonnet,Cartan-pr-curv,Chern-Deform-Pr-Curv}). 
Some global results were obtained by Lawson
and Tribuzy, and by Ros (see \cite{trib_and_lawson} and 
\cite{ARoss}). 

The methods used in \cite{Bonnet,Cartan-pr-curv}, and
\cite{Chern-Deform-Pr-Curv} are very powerful and beautiful but cannot
be used to study nonconstant mean curvature surfaces with umbilic
points. Until recently it was unknown whether such 
Bonnet mates exist. In particular, it was unknown
whether there are any Bonnet mates which are not included in the
Bonnet-Cartan-Chern classification. A general construction which
yields all immersions, with or without umbilic points, that admit
Bonnet mates was found in \cite{pipeka}. This construction made it
possible to prove that there exist infinitely many new Bonnet 
immersions, that is, Bonnet mates which have umbilic points and 
whose mean curvature is not constant in a neighborhood of the
umbilic points (\cite{saopaulo} and \cite{hqd}). 

This section contains results on the properties of the umbilic
points of Bonnet mates and on the the following rigidity
conjecture: {\it Let $(M,g)$ be a closed oriented Riemann surface
  and let $H:M\rightarrow\bR$ be an arbitrary function.
%  Then up to rigid motions, there exists at most one isometric
%  immersion $F:(M,g)\!\rightarrow \bR^3$ with mean curvature function
%  $H$}. 
Then up to rigid motions, there exists at most one isometric
immersion of $(M,g)$ into $\bR^3$ with mean curvature function
$H$}.

Throughout this note $(M, g)$ is an oriented, connected Riemann
surface; $F_1$, $F_2$ are isometric immersions of $(M, g)$ in
$\bR^3$ ; $\II_i$, $N_i$, and $H_i$ denote the second
fundamental form, the field of unit normals, and
the mean curvature function of the immersion $F_i$.
The mean curvatures of two isometric immersions $F_1$ and $F_2$
coincide if and only if $F_1$ and $F_2$ have the same 
principal curvatures at every point, and so if $H_1 = H_2$, then $p$ is an
umbilic for one of the immersions precisely if it is an umbilic for the 
other.

It is convenient to introduce the shape distortion tensor, 
$D = \II_1 - \II_{2}$, associated with 
$F_1$ and $F_2$. Thus, Bonnet's fundamental theorem of 
surface theory implies that $F_1 \sim F_2$ if and only if $D \equiv 0$. 
The Mainardi-Codazzi equations imply the following observation:
\begin{obs}\label{main-th-1} If the mean curvature functions 
of $F_1$ and $F_2$ coincide, then the associated shape distortion
operator is trace-free and divergence-free, i.e, 
its $(2,0)$ part, $D^{2,0}$, is holomorphic. 
\end{obs}
Observation \ref{main-th-1}  implies the following unique continuation 
property: 
\begin{coro} \label{unique-cont} Let $F_1: M\rightarrow \bR^3$ and
  $F_2: M\rightarrow \bR^3$ be isometric embeddings which have the same
  principal curvatures. If $F_1 \sim F_2$ on a nonempty open set, then
  $F_1 \sim F_2$ on $M$.
\end{coro}
Analyzing the  shape distortion tensor yields the following theorem describing the properties of the umbilic points of a Bonnet immersion.  
\begin{theorem} \label{umbilics} Suppose that $F_1$ and $F_2$ are Bonnet 
mates; then 

{\bf A.}  $D_p = 0$ if and only if $p$ is umbilic; moreover, 
for every umbilic point $p$  we have 
\[ \idx_{F_1}(p) = \idx_{D^{2,0}}(p) =  \idx_{F_2}(p) < 0,\]
where $\idx_{F_1}(p)$ and $\idx_{F_2}(p)$ denote the index of 
$p$ with respect to $F_1$ and $F_2$, respectively, and $\idx_{D^{2,0}}(p)$ 
is the index of $p$ with respect to the horizontal foliation of the 
quadratic differential $D^{2,0}$. (See  \cite{hopf} for the definitions of the 
different notions of index.) 

{\bf B.} Every umbilic point is a critical point of the mean and the 
Gauss curvatures. 
Furthermore, the trace-free part of $\II_{1}$ vanishes to a finite order 
at every umbilic point. 
\end{theorem}
According to Theorem \ref{umbilics}, if an isometric embedding, $F_1$,
admits a Bonnet mate, then the net formed by the curvature lines of $F_1$ 
has the same local character, that is, the same type of 
singularities as the net
formed by the horizontal and vertical foliations of a holomorphic
quadratic differential. Next we consider what can be said about an
embedding whose foliations of curvature lines are precisely the
horizontal (vertical) foliations of a holomorphic quadratic
differential.
\begin{theorem} \label{folio} Let $M$ be a closed oriented 
  Riemann surface,  and let %\newline
$F_{1}:(M,g)\rightarrow\bR^3$ be an isometric embedding 
whose    
net of curvature lines is the net formed 
by the principal stretch foliations of a holomorphic quadratic 
differential. Then an isometric embedding
  $F_2 : (M,g)\rightarrow\bR^3$ is congruent to $F_1$ if and
  only if $F_1\circ F_2^{-1}$ is orientation preserving and $H_1 =
  H_2$.
\end{theorem}
The proof of Theorem \ref{folio} also yields a theorem about immersions:
\begin{theorem} \label{foliimm} Let $M$ be a closed oriented 
  Riemann surface, and let %\newline
$F_{1}:(M,g)\rightarrow\bR^3$ be an isometric immersion whose    
net of curvature lines is the net formed 
by the principal stretch foliations of a holomorphic quadratic 
differential.  If $F_1$ admits a Bonnet mate, then 
its mean curvature is constant. 
\end{theorem}
In particular, the shape of most globally isothermic embeddings of a
closed surface is determined by the mean curvature. For example, if $M$
has genus one and $F_{1}$ is globally isothermic, or if $F_1(M)$ is a
surface of revolution, then every other isometric embedding, $F_2$,
such that $H_1 = H_2$ must be congruent to $F_1$.

The unique continuation property implies the following theorem: 
\begin{theorem} \label{conformal} Let $F_1, F_2 : (M, g) \rightarrow
\bR^3$ be two isometric embeddings, then $F_1 \sim F_2$ if and only if the
following set of conditions is satisfied: (i) $F_1 \circ F_2^{-1}$ is
orientation preserving; (ii) $H_1 = H_2$; (iii) there exists a 
conformal map $\varphi : S^2 \rightarrow S^2$ such that $N_1 = \varphi\circ N_2$. 
\end{theorem}
The unique continuation also yields a 
generalization of the following classical result: {\it If $H_1 = H_2$ and 
$N_1 = N_2$, then $F_1 = %\mbox
\text{translation} \circ F_2$}; see 
\cite{Gardner1969}. 
\begin{theorem} Let $F_1, F_2 : (M, g) \rightarrow
\bR^3$ be two isometric embeddings, then $F_1 \sim F_2$ if and only if the
following set of conditions is satisfied: (i) $F_1 \circ F_2^{-1}$ is
orientation preserving; (ii) $H_1 = H_2$; (iii) $(N_1 - N_2)(M)$
lies in a half-space.  
\end{theorem}
\section{Dirac spinors and conformal immersions}
To obtain deeper rigidity results and to find and classify 
possible counterexamples to the rigidity conjecture one needs a new
method for constructing surfaces.
Such a theory is outlined here and then used to investigate 
the following modified rigidity conjecture:

\medskip
{\it Let $(M,g)$ be a closed and oriented Riemann surface. Given a function $H$ on $(M, g)$, and a regular homotopy class,
  ${\cal F}$, of immersions of $M$ into $\bR^3$, there exists at most one,
up to rigid
  motions,  isometric
  immersion $F\in{\cal F}$ with mean curvature function $H$}. 
\medskip

Note that if the modified conjecture is false, then the rigidity
conjecture from the previous section is also false.

Instead of a new method for constructing surfaces one could try to use
the Weierstrass-Kenmotsu representation \cite{Kenmotsu79}.
To apply this representation one must solve Kenmotsu's system of differential equations satisfied 
by the differential $dF$ of a conformal immersion  $F$ with prescribed mean
curvature function $H$. The main difficulty is
that the system is nonhomogeneous, second order, and nonlinear if $H$
is nonconstant, and, in addition, the system is degenerate at points at
which $H=0$.  

The Weierstrass representation of minimal surfaces was reformulated in terms
of spinors in \cite{Sullivan89}.  These ideas were used in
\cite{KusnerSchmidt}, where the authors also indicated that there should be a
theory for general, not necessarily minimal, surfaces.  Indeed, such theories
were developed in \cite{Bobenko}, and later in \cite{kon} (see also
\cite{Taimanov}), and \cite{Jorg}. During the academic year 1995--1996 the
GANG seminar at the University of Massachusetts set out to investigate the
role of spinors in the geometry of immersed surfaces and to develop a general
theory of the spinor representation of surfaces.  The goal was to design an
efficient and useful calculus, and to give a transparent explanation of the
objects involved in the representation \cite{ulrich,franz,saopaulo,GANG}.

From now on $M$ always denotes a Riemann surface, that is, an oriented
surface with a chosen conformal structure. In particular, $M$ comes equipped
with a maximal holomorphic atlas $\displaystyle{\{(U_\alpha, z_\alpha)\}}$.
The conformal structure allows us to make sense of nonnegative two-forms,
and their square roots, the nonnegative and the nonpositive half-densities.
The square roots of a nonnegative two-form $wdx_\alpha \wedge
dy_\alpha$ defined by $\pm\sqrt{w dx_\alpha \wedge dy_\alpha } = \pm\sqrt{w}
|dz_\alpha|$ are sections in the fiber bundle $\hD$ of half-densities.  Thus,
the square root of a nonnegative two-form, in particular an area element, is
a half-density, and vice versa, the square of a half-density is
a two-form. Half-densities are necessary to introduce a conformally invariant
notion of surface tension:
\begin{definition} Let $F$ be a conformal immersion of $M$ into $\bR^3$ 
inducing the area element  $dA$ by pulling back the 
Euclidean metric by $F$, and let $H$ be the mean curvature function of the 
immersion $F$. The 
half-density $H\sqrt{dA}$ is called the mean curvature half-density of $F$.
\end{definition}
A conformal immersion, $F$, of a Riemann surface into 
$\bR^3$ defines a spin structure on the Riemann surface. The spin
structure characterizes uniquely the regular homotopy class of the
immersion. See \cite{Ulrich85} and \cite{Sullivan89}.  
The square root, $\sqrt{dF}$,
of the differential of $F$ is a section in the associated spinor
bundle, $\Sigma$.

For the rest of this paper we identify $\bR^{3}$ with the imaginary 
quaternions $\imH$. A spinor bundle on a Riemann surface is a quaternionic 
line bundle $\Sigma$ with transition functions 
$k_{\beta\alpha}:U_\alpha\cap U_\beta \rightarrow \C$ such that 
$k_{\beta\alpha}^2 = \partial z_\alpha/\partial z_\beta$.  
A choice of a spinor bundle is equivalent to choosing 
a square root bundle of the bundle of conformal $\bR^{3}$-valued 
one-forms on $M$. Indeed, the following theorem is known:
\begin{theorem}  \label{suff} A spinor bundle on a Riemann surface 
$M$ is a quaternionic line bundle $\Sigma$ on $M$ with a chosen 
endomorphism $J\in \Endo{\qH}(\Sigma)$ and a nontrivial 
quaternionic-hermitian,  fiber-preserving pairing 
\[(\cdot, \cdot): \Sigma \times \Sigma \rightarrow T^{*}M\otimes\qH,  \]
so that $J^{2} = -1$, 
and for every two spinors $\psi, \phi \in \Sigma$ based at the same
point $p$ we have 
\begin{equation} \label{jconfprop}
(\phi, \psi)(JX) = (J\phi, \psi)(X) =(\phi, J\psi)(X)
\end{equation}
for every $X\in T_{p}M$. Here $JX$ denotes the action of the complex 
structure on the vector $X$. 
\end{theorem}
Note that for every spinor $\psi\in\Sigma$ the form  $(\psi,\psi)=\omega$ is 
imaginary quaternionic valued. Furthermore, $(\psi,\psi)$ is a conformal 
$\bR^3 = \imH$ 
valued one-form on $M$. 
The spinor $\psi$ is interpreted as the square
root of $\omega$. Every spinor $\psi$ defines a nonnegative 
half-density $|\psi|^{2} := \|(\psi,\psi)\|$,  
where $\|\cdot\|$ is the Euclidean norm in $\bR^3$. The half-density  
$|\psi|^{2}$ measures the relative dilation associated with the 
conformal form $(\psi,\psi)$. 
A choice of a spinor bundle is equivalent to a
choice of a spin structure, that is, a holomorphic square root of the
canonical bundle $T^{(1,0)}M^{*}$ of $M$.  Indeed, given a 
spinor bundle $\Sigma$, define the complex line bundle of 
{\em positive spinors}  by $\Sigma_{+} \stackrel{{\mathrm{def}}}{=} 
\setB{\sigma\in \Sigma \, | \, J\sigma = \sigma i}$. For every 
positive spinor $\sigma\in \Sigma_{+}$ we define the one-form 
$\sigma^{2} \stackrel{{\mathrm{def}}}{=}-k(\sigma, \sigma)$. From 
%\mref
\eqref{jconfprop} it follows that 
$\sigma^{2}(JX)=i\sigma^{2}(X) = \sigma^{2}(X)i$ for every vector 
$X$. Thus, for every positive spinor $\sigma$, the one-form
$\sigma^{2}$ is complex valued and belongs to the canonical bundle 
$T^{(1,0)}M^{*}$ of $M$. The map sending $\sigma\in \Sigma_+$ to 
$\sigma^{2}$  
is a quadratic map from $\Sigma_{+}$ onto the canonical bundle of $M$.
Thus every $\psi\in \Sigma_+$ defines a conformal one-form 
$\psi^{2}\in T^{(1,0)}M^*$ and a nonnegative half-density 
$|\psi|^2$. 

There is a canonical bi-additive fiberwise pairing $\Sigma \times (T^*M
\otimes_{R}\qH) \rightarrow \calD^{\frac{1}{2}} \otimes_{R} \Sigma$ which 
defines  conformal Clifford product between spinors and 
$\qH$-valued one-forms. 
The {\it conformal Dirac operator} is the unique local linear operator 
% $ D : \Gamma(\Sigma) \rightarrow\Gamma(\calD^{\frac{1}{2}}\otimes\Sigma) $
\[ D : \Gamma(\Sigma) \rightarrow\Gamma(\calD^{\frac{1}{2}}\otimes\Sigma) \]
satisfying the Leibniz rule and such that $D\sigma = 0$ if $\sigma$ is a
local section of $\Sigma_+$ whose square is a closed one-form, i.e., if
$d\sigma^2 = 0$.  (Compare with \cite{Hitchin1974} and \cite{Atyah71}.)
\begin{definition} A section,  $\psi\in \Gamma(\Sigma)$, is called 
  a Dirac spinor if $D\psi = U\otimes \psi$ for some $U\in \Gamma(\hD)$; we
  say that $\psi$ generates the half-density $U$.
\end{definition}
Dirac spinors represent the square roots of the differentials of conformal
immer\-sions:
\begin{theorem} \label{maintheorem}
  For every nonvanishing $\psi\in \Gamma(\Sigma)$ the one-form $(\psi,
  \psi)$ is closed if and only if $\psi$ is a Dirac spinor. If $M$ is simply
  connected and $\psi$ is a nonvanishing Dirac spinor, then
\begin{equation} \label{weirstrass} 
F = \int (\psi, \psi)
\end{equation}
is a conformal immersion with mean curvature half-density equal to the
half-density generated by $\psi$.  Vice versa, given an oriented
surface $M$,
every immersion $F$ of $M$ into $\bR^3$ defines a unique, up to
isomorphism, spinor bundle
$\Sigma$ on the surface $M$ equipped with the pull-back 
conformal structure and precisely two Dirac spinors, $\psi$ and $-\psi$, such
that $(\pm\psi, \pm\psi) = dF$. The half-density generated by $\pm\psi$
equals the mean curvature half-density of $F$. 
\end{theorem}
Note that in fact $\Sigma = M\times\qH$ and after identifying $\bR^3$ with 
$\imH$ we have $(\psi, \phi) = \overline{\psi}dF\phi$, where
$\psi$ and $\phi$ are spinors based at the same point on $M$.
I will denote by $[\Sigma]$ the regular homotopy class of an immersion
inducing the spinor bundle $\Sigma$. 

The standard elliptic theory and the representation %\mref
\eqref{weirstrass}
imply the following local  existence result: {\it Every half-density can be realized
  locally as the mean curvature half-density of a conformal immersion}.
Global existence and the Dirichlet problem are
discussed in \cite{diracexists} and \cite{dirichlet}.

The outlined theory provides new insight into Bonnet's 
problem. 
%In parti\-cular, in the structure of the space of pairs
%of Bonnet mates on a closed surface $M$. 
Given a metric $g$ on $M$ and a 
function $H\neq %\mbox
\text{const}$, there exists at most one, up to
rigid motions,  Bonnet pair $F_{\pm}$
of isometric immersions of $(M,g)$ into $\bR^3$ with mean curvatures  
 $H_{\pm}=H$ (\cite{trib_and_lawson}). I am interested in the space 
of {\bf geometrically distinct} pairs of
Bonnet mates on a given Riemann surface $M$ and within the 
same regular homotopy class. The space of geometrically distinct pairs
of 
Bonnet mates is the space of Bonnet pairs modulo 
the natural gauge group ${\cal G} = {\cal E}(3) \times {\cal E}(3) \times \bR^{+}$
acting on it. Here ${\cal E}(3)$ is the 
Euclidean group of rigid motions. Indeed, if we rotate one of the 
Bonnet mates in the pair $F_{\pm}$ we obtain another pair of Bonnet
mates which is geometrically identical with the original pair
$F_{\pm}$. Furthermore, for every positive 
number $r > 0$ the Bonnet pair $rF_{\pm}$ is simply the original pair 
$F_{\pm}$ observed at a different scale; thus, the pairs $F_{\pm}$ and 
$rF_{\pm}$ are not geometrically distinct. The 
action of the group ${\cal G}$ 
preserves the conformal class, the mean curvature half-density, and the
regular homotopy class. Let ${\cal B}(M, \Sigma, U)$ be 
the moduli space of  geometrically distinct
pairs of Bonnet mates defined on the Riemann surface $M$ which belong
to the regular homotopy class associated with the spinor bundle
$\Sigma$ and induce a given potential $U$. 

\begin{theorem} [\cite{diracexists}] \label{ker2} For every half-density $U$
and 
regular homotopy class $[\Sigma]$ on the closed
  Riemann surface $M$, the moduli space ${\cal B}(M, \Sigma, U)$ of
  geometrically distinct Bonnet mates is either empty or it is a
  disjoint union of isolated components. Each nonempty component is either a
  point, a line, or a four-dimensional ball.
\end{theorem} 

Note that $\dim_{\qH}\ker(D - U)$ is finite if $M$ is closed. 
If $\dim_{\qH}\ker(D - U)=1$, then ${\cal B}(M, \Sigma, U)$ is empty. Furthermore:

\begin{theorem} [\cite{diracexists}] \label{dim2} Suppose that $U$ is a
  half-density on the Riemann surface $M$; if the spinor bundle $\Sigma$ is
  such that $\dim_{\qH}\ker(D - U) = 2$, then precisely one of the
  following alternatives holds: %\newline

{\bf(a) } ${\cal B}(M, \Sigma, U) = \emptyset$. %\newline

{\bf (b) } ${\cal B}(M, \Sigma, U)$ is a point. %\newline

{\bf (c) } ${\cal B}(M, \Sigma, U)$ is homeomorphic to a line segment. %\newline

{\bf (d) } ${\cal B}(M, \Sigma, U)$ is homeomorphic to a
four-dimensional ball. %\newline

Furthermore,  every conformal immersion that belongs to the 
regular homotopy class $[\Sigma]$ and induces the mean curvature half-density 
$U$ admits at most one, up to rigid motions, Bonnet mate. 
\end{theorem}

Theorem~\ref{dim2} provides an extension of
the results from \cite{trib_and_lawson} including constant mean curvature 
immersions. 

\begin{corollary} \label{l-tr-ext} Let $(M,g)$ be an oriented, closed
  Riemann surface, and let $H$ be a function defined on
$M$, possibly constant. Every regular homotopy class
$[\Sigma]$ such that $\dim_{\qH}\ker(D - U) = 2$ contains at most 
two geometrically distinct isometric immersions of $(M, g)$ into 
$\bR^3$ with mean curvature function $H$. 
\end{corollary}

%\noindent {\bf Remark}
\begin{nremark}
  One can draw a parallel between Bonnet's rigidity 
conjecture and Pauli's exclusion principle by thinking of the Dirac spinor
$\psi$ of an immersion $f$ as a wave function. Indeed, the Dirac spinor
satisfies the equation $D\psi = U\psi$ and defines the 
half-density $|\psi|^2 = \sqrt{dA}$, where $dA$ is the area element induced
on the surface by the immersion $f$.  Two noncongruent immersions $f_{\pm}$
are Bonnet mates if and only if they are in the same conformal class and induce
the same area element and the same mean curvature half-density $U_{+} =
U_{-}$. Thus if we consider only Bonnet mates within the same regular
homotopy class, then two noncongruent immersions $f_{\pm}$ are Bonnet mates
if and only if their respective Dirac spinors $\psi_{\pm}$ generate the same
potential $U_{+} = U_{-}$ and the same half-density $|\psi_{+}|^2 =
|\psi_{-}|^2$.  Furthermore, it is natural to think that the Dirac spinors
$\psi$ and $\psi\alpha$, where $\alpha$ is a unit quaternion, represent the
wave functions of the same particle, that is, the same quantum state, with
respect to two different Euclidean reference frames in the ambient space
$\bR^3$. Hence the modified rigidity conjecture is equivalent to saying that
the quantum state is uniquely determined by the potential (energy) and the
position density.
\end{nremark}
\bibliographystyle{plain}
\begin{thebibliography}{Atyah71}

\bibitem[At]{Atyah71} 
M. Atiyah, {\em Riemann surfaces and spin structures}, 
Ann. Sci. Ec. Norm. Sup. {\bf 4} (1971), 47--62. 
\MR{44:3350}
\bibitem[Bob]{Bobenko}
A. I. Bobenko, {\em Surfaces in terms of 2 by 2 matrices}, 
Harmonic Maps and Integrable Systems (A. Froddy and J. Wood, eds.), 
Aspects of Mathematics, Vieweg, Braunschweig, 1994. 
\MR{95m:58047}
\bibitem[Bon]{Bonnet}
O. Bonnet, {\em M{\'e}moire sur la th{\'e}orie des surfaces applicables}, 
J. {\'E}c. Polyt. {\bf 42} (1867).
\bibitem[Car]{Cartan-pr-curv} 
{\'E}. Cartan, {\em Sur les couples de surfaces applicables avec 
conservation des courbures principales}, Bull. Sc. Math. {\bf 66} (1942),
55--72, 74--85. 
\MR{5:216e}
\bibitem[Che]{Chern-Deform-Pr-Curv}
S. S. Chern, {\em Deformations of surfaces preserving principal curvatures}, 
Differential Geometry and Complex Analysis: a Volume Dedicated to the Memory of {H}. {E}. 
{R}auch (I.~Chavel and H. M. Farkas, eds.), Springer-Verlag, 1985, pp. 155--163.
\MR{86h:53005}
%\bibitem[GANG]{GANG} 
%{\em Notes on spinors and the generalized Weierstrass representation},
%notes based on the lectures given by G. Kamberov, P. Norman, U. Pinkall, F. Pedit,
%J. Richter, N. Schmidt, and I. Taimanov, in preparation (see also [Pi96], [Pe96], and [Ka97]).
\bibitem[Gar]{Gardner1969} 
R. Gardner, {\em An integral formula for immersions in Euclidean space},
J. Diff. Geom. {\bf 3} (1969), 245--252.
\MR{40:7992}
\bibitem[Hit]{Hitchin1974} 
N. Hitchin, {\em Harmonic spinors}, Adv. in Math. {\bf 14} (1974), 1--55.
\MR{50:11332}
\bibitem[Hopf]{hopf} {H.} Hopf, {\em Differential geometry in the large}, 
Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin and New York,
1983.
\MR{85b:53001}
%\bibitem[Ka]{kamberov-bonnet-96}
%G. Kamberov, {\em Recovering the Shape of a Surface 
%from the Mean Curvature}, to appear in {\em Proc. of the Intl. Conf. of
%Differential Equations and Dynamical Systems}, 1996, publ. by Marcel Dekker, 
%(1998). 
\bibitem[Ka]{kamberov-bonnet-96}
G. Kamberov, {\em Recovering the shape of a surface 
from the mean curvature}, to appear in {\em Proc. of the Intl. Conf. on 
Differential Equations and Dynamical Systems}, 1996, publ. by Marcel Dekker, 
(1998). 
\bibitem[Ka1]{saopaulo} {G.} Kamberov, 
{\em Lectures on quaternions, spinors, and surfaces}, 
Notes from the lectures at the University of S\~{a}o Paulo, Brazil, May--June,
1997.
\bibitem[Ka2]{hqd} {G.} Kamberov, {\em Quadratic differentials,
    quaternionic forms, and surfaces}, preprint. 
\bibitem[Ka3]{diracexists}
G. Kamberov, {\em Immersions of closed surfaces with 
prescribed mean curvature half-density}, in preparation. 
\bibitem[Ka4]{dirichlet}
G. Kamberov, {\em The Dirichlet problem for Dirac spinors}, preprint. 
\bibitem[Ken]{Kenmotsu79} {K.} Kenmotsu, 
{\em Weierstrass formula for surfaces of prescribed
  mean curvature}, Math. Ann. {\bf 245} (1979), 89--99. 
\MR{81c:53005b}
\bibitem[KNPP]{GANG} {G.} Kamberov, {P.} Norman, {F.} Pedit, {U.} Pinkall,
{\em Surfaces, quaternions, and spinors}, preprint.   
\bibitem[KPP]{pipeka}
{G.} Kamberov,  {F.} Pedit,  and {U.} Pinkall, {\em Bonnet pairs and isothermic 
surfaces}, 
GANG Preprint IV.21, 1996, to appear in Duke Math. J. {\bf 92} (1998). 
\bibitem[Ko]{kon}  {B. G.} Konopelchenko, {\em Induced surfaces and their 
integrable dynamics}, 
Stud. Appl. Math. {\bf 96} (1996),  9--51. 
\MR{96i:53011}
\bibitem[KT]{Taimanov}
{B. G.} Konopelchenko and {I. A.} Taimanov, 
{\em Generalized Weierstrass  formulae, soliton equations and 
                  the Willmore surfaces, I. Tori of revolution and
                  the mKDV}, 
preprint no. 187, Ruhr-Universit\"{a}t-Bochum, Fakult\"{a}t 
f\"{u}r Mathematik, 
1995. 
\bibitem[KS]{KusnerSchmidt}
{R.} Kusner and {N.} Schmidt, {\em The spinor representation of minimal 
surfaces}, 
GANG Preprint III.27, 1993.  Revised version, GANG Preprint 4.18. 
\bibitem[LT]{trib_and_lawson}
B. Lawson and R. Tribuzy, {\em On the mean curvature function for 
compact surfaces}, 
J. Diff. Geom. {\bf 16} (1981), 179--183.
\MR{83e:53060}
\bibitem[Pe]{franz} {F.} Pedit, {\em Lectures at the Summer School in 
Differential Geometry}, Durham, UK, Summer 1996. 
\bibitem[Pi]{Ulrich85}
{U.} Pinkall, {\em Regular homotopy classes of immersed surfaces}, 
Topology {\bf 24} (1985), 421--435. 
\MR{87e:57028}
\bibitem[Pi1]{ulrich} {U.} Pinkall, {\em Lectures at TU Berlin}, Spring 1996. 
\bibitem[Ri]{Jorg}
{J.} Richter, {\em Surfaces in terms of spinors}, 
Master Thesis, TU Berlin, 1995. 
\bibitem[Ros]{ARoss} {A.} Ros, 
{\em Compact hypersurfaces with constant scalar curvature and a
  congruence theorem}, J. Diff. Geom. {\bf 27} (1988), 215--223.
\MR{89b:53096}
\bibitem[Sul]{Sullivan89} 
D. Sullivan, {\em The spinor representation for minimal surfaces in space}, 
Unpublished notes, 1989. 
\end{thebibliography}

\end{document}



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