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\title{CHARACTERIZATIONS OF THE MEAN CURVATURE  AND A PROBLEM OF
G. CIMMINO\footnote{1991 {\em Mathematics Subject Classification.} Primary
49Q05, 53A05, 53A10; Secondary 35J60, 53C42.}}  \author{Johannes \, C.C. \,
Nitsche}  \date{University of Minnesota}
\maketitle

\vspace{.4in}


\section{}
Let ${\cal T}$ be a differential geometric surface 
immersed in Euclidean space $R^{3}$, $\pv$ an interior point of
${\cal T}$ and ${\cal S}_{\pv}(r)$ the sphere of radius $r$ centered at $\pv$.
For sufficiently small values of $r$, ${\cal S}_{\pv}(r)$ intersects ${\cal T}$
in a simple closed curve  ${\cal C}_{\pv}(r)$ dividing ${\cal S}_{\pv}(r)$ into
two parts. Denote by $\sigma_{\pv}(r) \cdot r^{2}$ the area of that part which is
penetrated by the normal vector of ${\cal T}$ at $\pv$. $\sigma_{\pv}(r)$ is
approximately equal to $2 \pi$. If ${\cal T}$ is a plane, then obviously
$\sigma_{\pv}(r) \equiv 2 \pi$. One may ask, as Gianfranco Cimmino did 
sixty-two years ago, to find all surfaces which share this property with the
plane  (\cite{cim/1932}, p. 288, \cite{nit/1975}, \S \,  875 ,
\cite{nit/1989}, A7.3.):

\vspace{.25in}

{\bf Problem :} To determine the class of all surfaces ${\cal T}$ which have 
the
property that for every interior point $\pv \in {\cal T}$ there is a number
$r_{\pv} > 0$ such that $\sigma_{\pv}(r) = 2 \pi$ for $0 < r < r_{\pv}$.

\vspace{.25in}

The interest in this problem is heightened by the observation that the right
helicoid is a member of this class, a fact which is easily confirmed. We shall
prove below that there are no others:

\vspace{.25in}

{\bf Theorem:} The plane and the right helicoid are the only differential
geometric surfaces with the property formulated in the Problem
above\footnote{The plane may be regarded as a degenerated helicoid of pitch 0.}.

\vspace{.25in}

One is invariably  reminded of another question, due to R. Osserman and not
yet settled, in which, surely by sheer coincidence, plane and right helicoid are
believed to be the sole possibilities\footnote{Of course, we have E. Catalan's
old discovery \cite{cat/1842} of the right helicoid as the only skew ruled
surface. For references to the literature see \cite{nit/1989}, \S \, 79.
Further characterizations of the right  helicoid can be found in
\cite{nit/1975}, \S \S \, 712, 726, 730 and in section {\bf 8} below.}; see e.g.
\cite{xav/1993}.

\vspace{.30in}

\section{}
A promising approach to our problem will suggest itself if we take note of
certain nontraditional characterizations\footnote{i.e. characterizations not
mentioned in the standard texts on differential geometry.} of the mean curvature
$H_{\pv}$ of ${\cal T}$ at $\pv$ that rely on the relation between the sphere
${\cal S}_{\pv}(r)$ and the curve ${\cal C}_{\pv}(r)$ and $H_{\pv}$. As we
mention these characterizations, a few general comments seem to be apposite.

 The
name mean curvature (courbure moyenne) for the arithmetic mean of the principal
curvatures $k_{1} = 1/R_{1}$ and $k_{2} = 1/R_{2}$ of ${\cal T}$, as well as a
discourse supporting its relevance, had been put forth 
 by Sophie Germain
(\cite{ger/1831}, p. 7ff). Earlier, more awkward descriptions were in use; for
instance, a minimal surface, i.e. a surface of vanishing mean curvature, was
said to be `a surface for which the principal curvature radii are equal and of
opposite sign.' 
 
 Let us call $l(r)$ the length of the curve ${\em C}_{\pv}(r)$ and
$L(r)$ the length of its spherical image. R. Sturm, an ahherent of J. Steiner's
and proponent of synthetic geometry a century ago, stated in \cite{stu/1883} that
the quotient $L(r)/l(r)$ converges for $r \rightarrow 0$ to the suitably oriented
mean curvature $H_{\pv}$ of ${\cal T}$ at $\pv$\footnote{Intriguing as Sturms
statement is -- he proves it with the tools of synthetic geometry in the spirit
of J. Steiner -- it appears to be incorrect.}. The  similarity
to O. Rodrigues's characterization in 1815 of the Gauss curvature
$k{_1} k_{2}$ by comparing a surface element with its
spherical image\footnote{Rodrigues missed, however, the {\em theorema egregium}
that Gauss obtained in its full generality in the course of 1826.} leads Sturm
to the following comment\footnote{Translation of the quotes are by the author.}
(\cite{stu/1883}, p. 382): \begin{quotation}
 ``So it can be seen that Sophie Germain's
curvature measure $\frac{1}{R_{1}} \, + \, \frac{1}{R_{2}}$ should not be
totally disparaged vis-\`{a}-vis that of Gauss, and that its definition is not
as arbitrary as it is for instance called by B. Cournot in his {\em Theory of
Functions}. To be sure, the curvature measure of Gauss will remain absolutely
the more valuable, and, in particular, the contemplations put forth by Miss
Germain in support of her propostion are hardly acceptable.


In any event, the mean curvature has found its due significance also in the
theory of minimal surfaces."
\end{quotation}
Sturm might have mentioned as well the eminent physical significance of the 
 quantity $H$ which was recognized in 1805 and 1806 by T. Young \cite{you/1805}
and P.S. Laplace \cite{lap/1839}, respectively, in their celebrated
investigations concerning the rise of a liquid in a capillary tube: the
pressure difference near an interface is proportional to the mean curvature of
the interface at the point under consideration.

Cournot's remark to which Sturm alludes is as follows (\cite{cou/1857}, p.
474): 
\begin{quotation} 
``Miss Sophie Germain has in her investigations
concerning elastic surfaces  and, above all, in the memoir (\cite{ger/1831}),
the final work of this remarkable woman, taxed her energies to
establish the fact that the curvature of a surface has as measure the function
$\frac{1}{R_{1}} \, + \, \frac{1}{R_{2}}$; but her arguments do not convince
at all. Very recently, the same idea has been taken up again by a young
professor, Mr. Renard, in a thesis defended before the Faculty of Sciences of
Paris, and that is known to us only through an abstract. It is only by analogy
that one can pass from the  definition for the curvature of a curve,
which is generally unchallenged, to the definition for the curvature of a
surface; and in geometry as well,  analogy  suggests itself rather than proves
itself. The best definition for the curvature of a surface will be
that which allows us to link the  largest possible number of surface properties
by means of the simplest possible expression." 
\end{quotation}

The above quotes are
examples of the strong (and strange) sentiments, to the point of proselytizing,
 that earlier geometers harbored concerning the value of various curvature
measures for surfaces,
 in particular the relative merits of the total (now called Gaussian) curvature
$K$ over the mean curvature $H$. But other curvature measures, to complement
Gaussian and mean curvature, have been proposed as well. For example, in 1889 F.
Casorati made the case for the measure $C = (k_{1}^{2} + k_{2}^{2})/2$
 (the letter $C$ presumably standing for Casorati). We have $2H^{2} = K + C$. In
fact, Casorati puts forth a fervent plea (\cite{cas/1890}, p. 109) recommending
to the aspiring young mathematicians that they pursue research dedicated to the
new curvature measure and urging the authors of textbooks to accord it a proper
place in their work\footnote{If one surveys the recent literature concerning
variational problems for curvature functions (see e.g. \cite{wil/1982}),  then
one realizes that today's mathematicians have heeded his plea; to be sure, they
have forgotten Casorati.}.
 One of his arguments was the fact that only the vanishing of
$C$, but not that of $K$ or $H$, characterizes the plane. Bianchi
(\cite{bia/1927}, p. 197) rejects this argument as unsound pointing out that it
could be continued ad infinitum: to the curvature measure $k_{1}^{4} + k_{2}^{4}$
etc.\footnote{In recent years, this expression has been discussed in connection
with engineering applications; for references see \cite{nit/1993}.}. Also of
interest in this connection is Darboux's balanced assessment (\cite{dar/1915},
sec. 497):
\begin{quotation}
``Memoirs have been written in search of the question which of the two
quantities [$k_{1} k_{2}$ or $(k_{1} + k_{2})/2$] should serve as measure for the
curvature of a surface at a given point ... ."
\end{quotation} 
Of course, the expression $H^{2}$ and the variational problem
\begin{equation}
\delta \int \int_{{\cal T}} H^{2} \, {\rm d} A  \label{var} = 0  \label{H}
\end{equation}
had been introduced and investigated by S.D. Poisson \cite{poi/1812} and S.
Germain \cite{ger/1821} in their investigations of problems in the theory of
elasticity\footnote{For appraisals of these contributions by experts in the
theory of elasticity, see e.g. \cite{nit/1991}, p. 83.}.


 Another characterization of the mean curvature in a similar vein as Sturm's is
due to G.Cimmino \cite{cim/1932}: 
\begin{equation}
H_{\pv} = \frac{1}{\pi} \, \lim_{r \rightarrow 0} \frac{2 \pi -
\sigma_{\pv}(r)}{r} \, \, \, .     \label{cim}
\end{equation}

It should be mentioned that the existence of the limit in (\ref{cim}) does
not require much of ${\cal T}$ by way of regularity. Thus, relation (\ref{cim})
is suitable for the definition of a generalized mean curvature. 
 If the surface ${\cal T}$ has a representation of class $C^{2}$,
however, it follows from (\ref{cim}) that the only candidates for the specified
class of surfaces are minimal surfaces. Whether a surface with vanishing
generalized mean curvature everywhere must be a minimal surface in the
classical sense is the other question raised by Cimmino in \cite{cim/1932}.

\vspace{.30in}

\section{}
In what follows we shall work with differential geometric surfaces in the sense
of \cite{nit/1989}, \S \, 47; the objects of our investigation can
then in fact be assumed to be surfaces of the regularity class $C^{\omega}$. It
will be necessary to find an expansion for the expression on the right hand
side of (\ref{cim}) in the form
\begin{equation}
\frac{2 \pi - \sigma_{\pv}(r)}{2 \pi r} = \sum_{n=0}^{\infty} \sigma^{(n)}
r^{n} \, \, , \label{cim1}
\end{equation}
so that $\sigma^{(0)} = H_{\pv}/2$. It had already been shown in 
\cite{nit/1975}, \cite{nit/1989}, \S \, 64 that
\begin{eqnarray}
\sigma^{(1)} & = & 0 \, \, , \label{cim2} \\
\sigma^{(2)} & = & \frac{1}{32} [{\bf \Delta} H + 2 H (H^{2} - K)]_{\pv}
\, \, . \label{cim3}
\end{eqnarray}
Here ${\bf \Delta}$ denotes the Beltrami operator of the surface ${\cal T}$.
For details, see section {\bf 4} below. We are then led to the following
extension of the limit relation (\ref{cim}):
\begin{equation}
[{\bf \Delta} H + 2 H (H^{2} - K)]_{\pv} = \frac{16}{\pi} \, \lim_{r
\rightarrow 0} \frac{2 \pi - \sigma_{\pv}(r) - r \pi H_{\pv}}{r^{3}} \, \, .
\label{cimm}
\end{equation}

The appearance of the differential operator  in (\ref{cim3}), (\ref{cimm}) is
challenging:  the differential equation ${\bf \Delta} H + 2 H (H^{2} - K)
= 0$ , a fourth order equation for the position vector of ${\cal T}$, is the
Euler-Lagrange equation for the variational problem (\ref{var}). 
Its nonparametric form had been derived by S.D. Poisson; see \cite{poi/1812}, pp.
224, 215. Its invariant form is due to W. Schadow; see \cite{tho/1924}, p. 56.
The differential equation vanishes, as does the limit (\ref{cimm}), for all
extremal surfaces of the variational problem (\ref{var}), in particular for all
Willmore surfaces, i.e. compact  differential geometric extremal surfaces
without boundary immersed in $R^{3}$; see \cite{wil/1982}, p. 131.

Of, course, the term $\sigma^{(2)}$ vanishes automatically for minimal surfaces.
Thus we will have to turn our attention to the higher terms, at least to
$\sigma^{(4)}$, as it can be seen without difficulty that $\sigma^{(3)} = 0$.

\vspace{.30in}

\section{}
Let ${\cal T}$ be a nonplanar  surface of class $C^{\omega}$ and let us consider
the neigborhood of a nonumbilic point on ${\cal T}$ (the umbilic points on a
nonplanar minimal surface are isolated; see \cite{nit/1975} , \cite{nit/1989},
\S \, 125). It  shall be assumed that this point  is the origin of our coordinate
system and that the surface is locally represented in nonparametric form
 $z = f(x,y)$ 
 satisfying the conditions $f(0,0) = f_{x}(0,0) = f_{y}(0,0) = f_{xy}(0,0) =
0$. Then 
\begin{equation}
f(x,y) = \sum_{n=2}^{\infty} P_{n}(x,y) \, \, , \, \, P_{n}(x,y) =
\sum_{m=0}^{n} a_{n-m,m} x^{n-m} y^{m} \, \, , \label{exp}
\end{equation}
with $P_{2}(x,y) = a_{2,0} x^{2} + a_{0,2} y^{2}$. (For a minimal surface, we
have obviously $a_{2,0} + a_{0,2} = 0$, but $a_{2,0} \neq 0$.)

 Using
spherical coordinates $x = r \sin \vartheta \cos \varphi \, , \, y = r \sin
\vartheta \sin \varphi \, , \, z = r \cos \vartheta$, we set
\begin{eqnarray*}
P_{n}(x,y) & = & r^{n} \sin^{n}\vartheta \sum_{m=0}^{n} a_{m-n,m}
\cos^{n-m}\varphi \sin^{m}\varphi \\ 
 & = & r^{n} \sin^{n}\vartheta \,  p_{n} (\varphi) \, \, ,
\end{eqnarray*}
The curve ${\cal C}_{\ov}(r)$ is given by the equation 
$r \cos \vartheta = f(r \sin \vartheta \cos
\varphi , r \sin \vartheta \sin \varphi) \, \, .
$
Thus, for small values of $r$, the angle $\vartheta$ is nearly equal to $\pi/2$
and the equation for ${\cal C}_{\ov}(r)$ can be solved in the form 
$\vartheta
= \vartheta(r,\varphi)$; we also set $\alpha(r,\varphi) = \pi/2 -
\vartheta(r,\varphi)$. Then
\[
\sin \alpha(r,\varphi) = \sum_{n=2}^{\infty} r^{n-1} p_{n}(\varphi)
\cos^{n}\alpha \, \, , 
\] 
so that
\[ 
\sin \alpha (r,\varphi) = r q_{2}(\varphi) + r^{2} q_{3}(\varphi) + \ldots
\, \, .
\] 
The first coefficient functions $q_{n} = q_{n}(\varphi)$ are
\begin{eqnarray*}
q_{2} & = & p_{2} \, \, , \\
q_{3} & = & p_{3} \, \, , \\
q_{4} & = & p_{4} - p_{2}^{3} \, \, , \\
q_{5} & = & p_{5} - \frac{7}{2} p_{2}^{2} p_{3} \, \, , \\
q_{6} & = & p_{6} - 4 p_{2} p_{3}^{2} - 4 p_{2}^{2} p_{4} + 2 p_{2}^5 \, \, .
\end{eqnarray*}
Now
\[
\sigma_{\ov}(r) \cdot r^{2} = r^{2} \int_{0}^{2 \pi} {\rm d} \varphi 
\int_{0}^{\vartheta(r,\varphi)} {\rm d} \vartheta \, \sin \vartheta = r^{2}
\int_{0}^{2 \pi} (1 - \cos \vartheta(r,\varphi) ) \, {\rm d} \varphi
\]
and
\[
 \sigma_{\ov}(r) = 2 \pi - \int_{0}^{2 \pi} \sin \alpha (r,\varphi) \, {\rm d}
\varphi \, \, .
\]
As a result we obtain the expansion (\ref{cim1}),
where
\[ 
\sigma^{(n)} = \frac{1}{2 \pi} \int_{0}^{2 \pi} q_{n+2}(\varphi) {\rm d} \varphi
\, \, .
\]
 
It is desirable to express the expansion coefficients $\sigma^{(n)}$ in terms
of geometric invariants of the surface ${\cal T}$. For this purpose, note that
the mean and Gaussian curvatures of ${\cal T}$ are
\begin{eqnarray*}
H & = & \frac{(1 + f_{y}^{2}) f_{xx} - 2 f_{x} f_{y} f_{x,y} + (1 +
f_{x}^{2})}{2 \sqrt{1 + f_{x}^{2} + f_{y}^{2}}^{3}} \, \, , \\
K & = & \frac{f_{xx} f_{yy} - f_{xy}^{2}}{(1 + f_{x}^{2} + f_{y}^{2})^{2}} \,
\, ,
\end{eqnarray*}
while the Beltrami operator applied to a function $\zeta(x,y)$ defined on
${\cal T}$ is given by 
\[
{\bf \Delta} \zeta = \frac{(1 + f_{y}^{2}) \zeta_{xx} - 2 f_{x} f_{y}
\zeta_{xy} + (1 + f_{x}^{2}) \zeta_{yy}}{1 + f_{x}^{2} + f_{y}^{2}} - \frac{2
H (f_{x} \zeta_{x} + f_{y} \zeta_{y})}{\sqrt{1 + f_{x}^{2} + f_{y}^{2}}} \, \,
;
\] 
see \cite{nit/1989}, \S \, 61.
It follows that
\begin{eqnarray*}
H_{\ov} & = & a_{2,0} + a_{0,2} \, \, ,\\
K_{\ov} & = & 4 a_{2,0} a_{0,2} \\
(H^{2} - K)_{\ov} & = & (a_{2,0} - a_{0,2})^{2} \, \, , \\
({\bf \Delta} H)_{\ov} & = & 4 \, [3a_{4,0} + a_{2,2} + 3a_{0,4}] \\
 & & + 4(a_{2,0}
+ a_{0,2}) \, [2a_{2,0} a_{0,2} - 3(a_{2,0}^{2} + a_{0,2}^{2}] \, \, .
\end{eqnarray*}
A computation then shows that that the first expansion terms $\sigma^{(0)} \, ,
\, \sigma^{(1)} \, , \, \sigma^{(2)}$ have the values already laid down in
(\ref{cim}), (\ref{cim2}), (\ref{cim3}) above. 

\vspace{.30in}

\section{}
For the determination of $\sigma^{(4)}$, we shall restrict ourselves here
to the
consideration of minimal surfaces, with the normalization $a_{2,0} = - a_{0,2}
=1$. The explicit expressions for the first homogeneous polynomials in the
expansion (\ref{exp}) are

\begin{eqnarray}
P_{2}(x,y) & = & x^{2} - y^{2} \, \, , \label{e2} \\
P_{3}(x,y) & = & {\bf Re} \{ (A_{3} - i B_{3}) (x + i y)^{3}) \} \, \, ,
\label{e3} \\ 
P_{4}(x,y) & = & {\bf Re} \{ (A_{4} - i B_{4}) (x + i y)^{4} \} +
\frac{2}{3} (x^{4} - y^{4}) \, \, , \label{e4} \\
P_{5}(x,y) & = & {\bf Re} \{ (A_{5} - i B_{5}) (x + i y)^{5} \} \nonumber\\
 & & + \frac{4}{5} A_{3} (3 x^{5} - 5 x y^{4}) + \frac{4}{5}  B_{3} (5 x^{4} y
- 3 y^{5} ) \, \, , \label{e5}\\
P_{6}(x,y) & = & {\bf Re} \{ (A_{6} - i A_{6}) ( x + i y)^{6} \} + \frac{4}{15}
A_{4} ( 11 x^{6} - 15 x^{4} y^{2} - 15 x^{2} y^{4} + 11 y^{6} ) \nonumber \\
& & + \frac{32}{5} B_{4} ( x^{5} y + x y^{5}  ) + \frac{1}{10} (33 A_{3}^{2} +
15 B_{3}^{2}) x^{6} + \frac{36}{5} A_{3} B_{3} (x^{5} y - x y^{5} ) \nonumber \\
& & + \frac{9}{2} (B_{3}^{2} - A_{3}^{2}) (x^{4} y^{2} + x^{2} y^{4} ) -
\frac{1}{10} (15 A_{3}^{2} + 33 B_{3}^{2}) y^{6} + \frac{32}{45} ( x^{6} -
y^{6}) \, \, \label{e6} .
\end{eqnarray}
The coefficients $A_{n} \, , \, B_{n}$ denote arbitarary real parameters. For
instance, in the case of Enneper's minimal surface we have $A_{3} = B_{3} = 0
\, , \, A_{4} = B_{4} = 0 \, , \, A_{5} = B_{5} = 0 \, , \, A_{6} = 1/15 \, ,
\, B_{6} = 0$; see \cite{nit/1975} , \cite{nit/1989}, \S \, 89. 

A lengthy computation shows that
\begin{equation}
\sigma^{(4)} = \frac{1}{3} \, A_{4} \, \, . \label{cim4}
\end{equation}
As a consequence, the requirement $\sigma^{(4)} = 0$ implies the condition
\begin{equation}
A_{4} = 0 \label{cim35}
\end{equation}
or the equivalent relations
\begin{equation}
f_{xxxx}(0,0) = - f_{yyyy}(0,0) = 16 \, \, , \, \, f_{xxyy}(0,0) = 0 \, \, .
\label{cim4'} 
\end{equation}

Note that condition (\ref{cim4}) is formulated here in terms of a local
coordinate system; it must be satisfied in {\em all}  nonumbilic points of
${\cal T}$. (In the case of Enneper's surface, as well as many other minimal
surfaces,  it turns out that condition (\ref{cim4}) holds only at the point of
highest symmetry.)

\vspace{.30in}

\section{}
For the geometric interpretation of condition (\ref{cim4}), the surface ${\cal
T}$ is assumed to be before us in parametric representation ${\cal T} = \{
\xv = \xv(u,v) \, ; \, (u,v) \in P \}$ , where the position vector $\xv(u,v) =
(x(u,v),y(u,v),z(u,v))$ is defined in a neigborhood $P$ of the nonumbilic
reference point in the $(u,v)$-parameter plane.
 The unit normal vector of ${\cal T}$ will be denoted by $\Xv(u,v) =
(X(u,v) , Y(u,v) , Z(u,v))$. We shall take $u$ and $v$ as curvature line
parameters which for a minimal surface are known to be isothermal; see
\cite{nit/1975}, \cite{nit/1989}, \S \, 172. The first fundamental form of
${\cal T}$ has the form $ds^{2} = E(u,v) (du^{2} + dv^{2})$, i.e. $E(u,v)
\equiv \xv_{u}^{2} = G(u,v) \equiv \xv_{v}^{2}$ and $F(u,v) \equiv \xv_{u}
\cdot \xv_{v} = 0$. The coefficients $L(u,v) , M(u,v) , N(u,v)$ of the second
fundamental form of ${\cal T}$ satisfy the conditions $L = -N = const \, ,
\, M = 0$. Without loss of generality, we may assume that $L(u,v) = 1$.

It is convenient to employ the complex variable $w = u + i v$ and the
complex-valued differentiations $\partial/\partial w = (\partial/\partial u -
i \, \partial/\partial v)/2$ and $\partial/\partial \overline{w} =
(\partial/\partial u + i \, \partial/\partial v)/2$. The fundamental equations
of differential geometry become
\begin{eqnarray}
\xv_{ww} & = & \frac{E_{w}}{E} \, \xv_{w} + \frac{1}{2} \, \Xv \, \, ,
\label{1} \\ 
\xv_{w \overline{w}} & = & 0 \, \, , \label{2} \\
\Xv_{w} & = & - \frac{1}{E} \, \xv_{\overline{w}} \, \, , \label{3}
\end{eqnarray}
while the theorema egregium is expressed by the equation
\begin{equation}
E_{w \overline{w}} = \frac{1}{E} \, |E_{w}|^{2} + \frac{1}{2}
\, \, \, \, {\rm or} \, \, \, \, \Delta  \log  E = \frac{2}{E} \, \,. \label{4}
\end{equation}

The position vector  of ${\cal T}$ is the real part of an analytic
vector ${\bf \Phi}(w) = (\phi_{1}(w), \phi_{2}(w), \phi_{3}(w))$ which satisfies
in P the relations 
\begin{equation}
{\bf \Phi}' \cdot {\bf \Phi}'  = 0 \, , \, |{\bf \Phi}'|^{2} = 2 E \, \,  . \label{5} 
\end{equation}
It follows from (\ref{1}) -- (\ref{5}) that 
\begin{equation}
{\bf \Phi}'
\cdot {\bf \Phi}'' = 0 \, , \, {\bf \Phi}' \cdot {\bf \Phi}''' = - {\bf \Phi}''
\cdot {\bf \Phi}'' = -1 \label{10}
\end{equation}
and
\begin{equation}
  \overline{{\bf \Phi}'} \cdot {\bf \Phi}'' = 2 E_{w} \, , \, \overline{{\bf
\Phi}'} \cdot {\bf \Phi}''' = 2 E_{ww} \, \, . \label{11a}
\end{equation}
 We also have the relations 
\begin{equation}
\Xv \cdot {\bf \Phi}'' =  1 \, , \, 
\Xv \cdot {\bf \Phi}''' =  \frac{E_{w}}{E}  \, , \, 
\Xv \cdot {\bf \Phi}^{iv} =  2 \frac{E_{ww}}{E} - \frac{E_{w}^{2}}{E^{2}} \,
\, . \label{11}
\end{equation} 

Let now be $w^{0} = u^{0} + i v^{0}$ be a nonumbilic point in $P$; we also
write $\xv(u^{0},v^{0}) = \xv^{0} \, , \, E(u^{0},v^{0}) = E^{0}$ etc. In a
neighborhood of this point, the surface ${\cal T}$ will be referred to a
local coordinate system with axes in the directions of the orthonormal unit
vectors 
\[
\ev_{1} = \frac{1}{\sqrt{E^{0}}} \, \xv_{u}^{0} \, , \,  \ev_{2}
= \frac{1}{\sqrt{E^{0}}} \, \xv_{v}^{0} \, , \, \ev_{3} = \Xv^{0} \, \, ,
\] 
and new coordinates $(\xi , \eta , \zeta)$ will be introduced according to
\begin{eqnarray*}
\xi & = & \frac{1}{2 E^{0}} \ev_{1} \cdot (\xv(u,v) - \xv^{0}) \, \, , \\
\eta & = & \frac{1}{2 E_{0}} \ev_{2} \cdot (\xv(u,v) - \xv^{0}) \, \, , \\
\zeta & = & \frac{1}{2 E^{0}} \ev_{3} \cdot (\xv(u,v) - \xv^{0}) \, \, .
\end{eqnarray*}
It is necessary to express the equation of ${\cal T}$ in the form
\[
\zeta = f(\xi,\eta) = \sum_{n=2}^{\infty} P_{n}(\xi,\eta) \, \, , \label{greek}
\]
as in (\ref{exp}). For this purpose, observe that $\xv_{u} + i \xv_{v} =
\overline{{\bf \Phi}'}$, so that
\[
\lambda \equiv 2 \sqrt{E^{0}} (\xi + i \eta) = \frac{1}{E^{0}} \overline{{\bf
\Phi}'}\cdot(\xv(u,v) - \xv^{0}) \, \, ,
\]
 while
\[
\xv(u,v) - \xv^{0} = {\bf Re} \{ (w - w^{0}) {\bf \Phi}'(w^{0}) + \frac{1}{2}
(w - w^{0})^{2} {\bf \Phi}''(w^{0}) + \ldots \} \, \, .
\]

A computation utilizing the relations (\ref{1}) -- (\ref{11}) leads to
\[
w - w^{0} =  \lambda - \frac{1}{2} \frac{E_{w}^{0}}{E^{0}} \lambda^{2} 
  + \{ \frac{1}{2} (\frac{E_{w}^{0}}{E^{0}})^{2} - \frac{1}{6}
\frac{E_{ww}^{0}}{E^{0}} \} \lambda^{3} + \frac{1}{12} \frac{1}{E^{0}}
\overline{\lambda}^{3} + \ldots
\]
and ultimately to the expansion (\ref{greek}), where, as in (\ref{e2}),
(\ref{e3}), (\ref{e4}),
\begin{eqnarray}
P_{2}(\xi,\eta) & = & \xi^{2} - \eta^{2} \, \, , \label{f2} \\
P_{3}(\xi,\eta) & = & {\bf Re} \{ (A_{3} - i B_{3}) (\xi + i \eta)^{3}) \} \, \,
, \label{f3} \\ 
P_{4}(\xi,\eta) & = & {\bf Re} \{ (A_{4} - i B_{4}) (\xi + i \eta)^{4} \} +
\frac{2}{3} (\xi^{4} - \eta^{4}) \, \, . \label{f4} 
\end{eqnarray}
Here
\begin{eqnarray}
A_{3} - i B_{3} & = & - \frac{4}{3} (\frac{E_{w}}{E^{3/2}})^{0} \, \, ,
\label{g3} \\
A_{4} - i B_{4} & = & - \frac{2}{3} ( E_{ww} - 5 \frac{E_{w}^{2}}{E})^{0} \\
& = & \frac{1}{6} \{ E^{5} (E^{-4})_{ww}) \}^{0} \, \, . \label{g4}
\end{eqnarray}

In view of condition (\ref{cim35}) and the remarks at the end of section {\bf 5},
we must have 
\begin{equation}
{\bf Re} \{ (E^{-4})_{ww} \} = 0 \, \, \, \, {\rm or} \, \, \, \, (E^{-4})_{uu} -
(E^{-4})_{vv} = 0 \label{h4}
\end{equation}
at {\em all}  nonumbilic points $w = u + i v$ in the domain $P$. This
differential equation joins the theorema egregium (\ref{4}) as condition for
the first fundamental form of our surface ${\cal T}$. Their implications will be
discussed in the next section.
 For this purpose, it is convenient to switch
from the parametrization of ${\cal T}$ by lines of curvature to the
parametrization by asymptotic lines, which is also an isothermal
parametrization; see \cite{nit/1975}, \cite{nit/1989}, \S \, 172.  It may be
assumed that the parameter domain $P$ contains the origin $u = v = 0$, but no
umbilic points of ${\cal T}$. Setting $p(u,v) = E^{-4}(u,v)$, our conditions
then become 
\begin{equation}
p(u,v) > 0 \, \, , \, \, p_{uv} = 0 \, \, , \, \, \Delta \log p + 8 p^{1/4} = 0
\label{i4}
\end{equation}
for all points $(u,v) \in P$. 

\vspace{.30in}

\section{}
We shall prove  the following

\vspace{.25in}

{\bf Lemma:} A function $p(u,v)$ satisfying conditions (\ref{i4}) depends only
on {\em one} of its variables, i.e. $p = p(u)$ or $p = p(v)$
.
\vspace{.25in}

It is a consequence of this lemma that the coefficient $E$ of the first
fundamental form of ${\cal T}$ is constant on each member of one of
the two families of asymptotic lines of ${\cal T}$.

Set $q(u,v) = p(u,v)/p(0,0)$. Then $q(0,0) = 1$ and $\Delta \log q = - k
q^{1/4} \, , \,  k = 8 p^{1/4}(0,0) > 0$. It is clear that $q(u,v)$ has the
form $q(u,v) = 1 + f(u) + g(v)$, where 
\begin{eqnarray*}
f(u) & = & a_{1} u + a_{2} u^{2} + \ldots \, \, , \\
g(v) & = & b_{1} v + b_{2} v^{2} + \ldots \, \, .
\end{eqnarray*}
Substitution of the expression for $q$ into (\ref{4}) yields
\begin{equation}
(1 + f + g) (f'' + g'') - f'^{2} - g'^{2} + k (1 + f + g)^{9/4} =
\sum_{m,n=0}^{\infty} c_{m,n} u^{m}v^{n} = 0 \, \, . \label{sep}
\end{equation}

Some of the first coefficients $c_{m,n}$ are as follows:
\begin{eqnarray}
c_{0,0} & = & 2(a_{2} + b_{2}) - a_{1}^{2} - b_{1}^{2} + k \, \, , \label{00} \\
c_{1,0} & = & 6a_{3} - 2a_{1} (a_{2} - b_{2}) + \frac{9}{4} k a_{1} \, \, ,
\label{10} \\ 
c_{0,1} & = & 6b_{3} + 2b_{1} (a_{2} - b_{2}) + \frac{9}{4} k b_{1}
\, \, , \label{01} \\
c_{2,0} & = & 12 a_{4} - 2 a_{2} (a_{a} - b_{2}) + \frac{9}{32} k (8 a_{2} + 5
a_{1}^{2}) \, \, , \label{20} \\
 c_{1,1} & = & 6 a_{1} b_{3} + 6 a_{3} b_{1} + \frac{45}{16}
k a_{1} b_{1} \, \, , \label{11c} \\
c_{0,2} & = & 12b_{4} + 2b_{2} (a_{2} - b_{2}) + \frac{9}{32} k (8b_{2} + 5
b_{1}^{2}) \, \, , \label{02} \\
c_{2,1} & = & 12 a_{4} b_{1} + 6 a_{2} b_{3} + \frac{45}{128} k (8 a_{2} b_{1}
+ a_{1}^{2} b_{1}) \, \, , \label{21} \\ 
c_{1,2} & = & 12a_{1}b_{4} + 6a_{3}b_{2}
+ \frac{45}{128} k (8a_{1}b_{2} + a_{1}b_{1}^{2}) \, \, . \label{12} 
\end{eqnarray}
 Combination of the equations (\ref{10}), (\ref{01}), (\ref{11c}), the
equations (\ref{01}), (\ref{20}), (\ref{21}) and the equations (\ref{10}),
(\ref{02}), (\ref{12}), respectively, leads to the three conditions
\[
a_{1} b_{1} = 0 \, \, , \, \, a_{1} (8 b_{2} + 5 b_{1}^{2}) = 0 \, \, , \, \,
b_{1} (8 a_{2} + 5 a_{1}^{2}) = 0 \, \, .
\] 
 
Assume first that $b_{1} = 0$. Then we must have $a_{1} b_{2} = 0$. Inspection of
(\ref{01}) also shows that  $b_{3} = 0$. 
  Two cases have to be considered:

\vspace{.2in}

{\bf Case 1:} $b_{2} = 0$. The differential equation (\ref{sep}) for the 
value $u = 0$ becomes
\begin{equation}
(1 + g) (2 a_{2} + g'') - a_{1}^{2} - g'^{2} + k (1 + g)^{9/4} = 0 \, \, .
\label{sep1}
\end{equation} 
Since $b_{1} = b_{2} = 0$ and, in view of (\ref{00}), $2 a_{2} - a_{1}^{2} + k =
0$, we see that $g(v) \equiv 0$ is the only solution of (\ref{sep1}). It
follows that $q(u,v) = 1 + f(u)$. 

\vspace{.2in}

{\bf Case 2:} $a_{1} = 0 \, , \, b_{2} \neq 0$. By a previous argument, also
$a_{3} = 0$. The coefficient $c_{2,2}$ becomes
\[
c_{2,2}  =  12 (a_{2} b_{4} + a_{4} b_{2}) + \frac{45}{16} k a_{2} b_{2} 
= - \frac{819}{16} k a_{2} b_{2} \, \, .
\]
The condition $c_{2,2} = 0$ implies  $a_{2} = 0$, and an argument similar to
the one used before shows now that $q(u,v) = 1 + g(v)$.

\vspace{.2in}

The initial assumption $a_{1} = 0$ leads to the same conclusion. The lemma is
proved.


\vspace{.30in}

\section{}
Consider a portion of the minimal surface ${\cal T}$ in
an isothermal parametrization by asymptotic lines and free of umbilic points.
 We know now that the
coefficient $E(u,v)$ of the first fundamental form depends only on one of its
two variables, say $E = E(u)$. The function $h(u) = \sqrt{E(u)} > 0$ satisfies
the differential equation
\begin{equation}
h h'' = 1 + h'^{2} \, \, . \label{the}
\end{equation}
The general solution of (\ref{the}) is
\begin{equation}
h(u) = a \cosh \, (\kappa u) + b \sinh \, (\kappa u) \, \, , \label{the1}
\end{equation}
where $\kappa$ and $b$ denote arbitrary constants and the  coefficient $a >
0$ satisfies the relation $a^{2} = b^{2} + 1/\kappa^{2}$. Integration of the
fundamental equations of differential geometry whose details are omitted here
leads to the following conclusion: In a suitably chosen $(y^{1}, y^{2},
y^{3})$-coordinate system, the equation of ${\cal T}$ takes the form $\{ \yv =
\yv(u,v)
 \,  ; \, (u,v) \in P \}$ where
\begin{equation}
\yv = (h'(u) \cos (\kappa v) , h'(u) \sin (\kappa v) , \lambda v) \, \, .
\label{sol}
\end{equation}
(\ref{sol}) shows that ${\cal T}$ is a right helicoid.With this, Cimmino's
question has been answered


 \vspace{.30in}

\begin{thebibliography}{9}

\bibitem{bia/1927}
L. Bianchi: {\em Lezioni di geometria differenziale,} 3rd ed., vol. 1, part 1.
Nicola Zanicelli, Bologna, 1914  .

\bibitem{cas/1890}
F. Casorati: Mesure de la courbure des surfaces suivant l'id\'{e}e commune.
Ses rapports avec les mesures de courbure Gaussienne et moyenne. {\em Acta
Math.} {\bf 14} (1890-1), 95-110.

\bibitem{cat/1842}
E. Catalan: Sur les surfaces r\'{e}gl\'{e}es dont l'aire est un minimum. {\em
J. Math\'{e}m. p. appl.} (1) {\bf 7} (1842), 203-211.


\bibitem{cim/1932}
G. Cimmino: Sulla curvatura media delle superficie. {\em Rend. Circ. Mat.
Palermo} {\bf 56} (1932), 281-288.

\bibitem{cou/1857}
B. Cournot: {\em Trait\'{e} \'{e}l\'{e}mentaire de la th\'{e}orie des fonctions
et du calcul infinit\'{e}simal,} Vol. 1. Librairie de L. Hachette , Paris, 1857.

\bibitem{dar/1915}
G. Darboux: {\em Le\c{c}ons sur la th\'{e}orie g\'{e}n\'{e}rale des surfaces et
les applications g\'{e}om\'{e}triques du calcul infinit\'{e}simal,} vol. 2. 
2nd edn, Gauthier-Villars, Paris, 1915.


\bibitem{ger/1821}
S. Germain: {\em Recherches sur la th\'{e}orie des surfaces \'{e}lastiques.} 
Imprimerie de Huzard-Courcier, Paris, 1821.

\bibitem{ger/1831}
--------------------: M\'{e}moire sur la courbure des surfaces. {\em J. r.
angew. Mathematik} {\bf 7} (1831), 1-29.

\bibitem{lap/1839}
P.S. Laplace: {\em Trait\'{e} de m\'{e}canique c\'{e}leste.} Suppl\'{e}ment aux
livre X. 1805 and 1806, resp. In {\em \OE uvres compl.} Vol. 4, Gauthier-Villars,
Paris. Annotated Engl. translation by N. Bowditch, 1839; reprinted by Chelsea,
New York, 1966.


\bibitem{nit/1975}
J.C.C. Nitsche: {\em Vorlesungen \"{u}ber
Minimalfl\"{a}chen.} Springer-Verlag,
Berlin--Heidelberg--New York, 1975.


\bibitem{nit/1989}
--------------------: {\em Lectures on minimal surfaces, Volume
I.} Cambridge University Press, Cambridge--New York--New
Rochelle--Melbourne--Sydney, 1989.

\bibitem{nit/1991}
--------------------: Periodic surfaces which are extremal for energy functionals
containing curvature functions.  IMA vol. in Math. and its Appl. {\bf 51}
(1993), 69-98.

\bibitem{nit/1993}
--------------------: Boundary value problems for variational integrals
involving surface curvatures. {\em Quarterly Appl. Math.} {\bf 51} (1993),
363-387.


\bibitem{poi/1812}
S.D. Poisson: M\'{e}moire sur les surfaces
\'{e}lastiques. {\em M\'{e}m. Cl. Sci. Math\'{e}m. Phys.
Inst. de France,} 1812, 2nd pt., 167-225.

\bibitem{poi/1833}
--------------------: M\'emoire sur le calcul des variations. {\em M\'{e}m. Acad.
Roy. Sci. Inst. de France} {\bf 12} (1833), 223-331.

\bibitem{stu/1883}
R. Sturm: Ein Analogon zu Gauss' Satz von der Kr\"{u}mmung der Fl\"{a}chen.
{\em Math. Ann.} {\bf 21} (1883), 379-384.


\bibitem{tho/1924}
G. Thomsen: \"{U}ber konforme Geometrie I. Grundlagen der konformen
Fl\"{a}chentheorie. {\em Abh. Math. Sem. Hamburg Univ.} {\bf 3} (1924), 31-56.


\bibitem{wil/1982}
T.J. Willmore: {\em Total curvature in Riemannian geometry.}
Ellis Horwood Ltd., Chichester, 1982.

\bibitem{xav/1993}
F. Xavier: On the structure of complete simply connected embedded minimal
surfaces. {\em J. Geom. Anal.} {\bf 3} (1993), 513-527.

\bibitem{you/1805}
T. Young: An essay on the cohesion of fluids. {\em Phil. Transact. Roy. Soc.
London} {\bf 95} (1805), 65-87.




\end{thebibliography}



\end{document}

