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\documentstyle[12pt,leqno]{article}

\def\BbR{{\rm I\!R}}
\newcommand {\Z}{ Z\!\!\!Z }
\begin{document}

\title{On the problem of realization of a given \\
Gaussian curvature function }
\author{V. Arnold$^1$}
\date{}

\footnotetext[1]{Partially supported by the Russian Basic Research Foundation,
project {\bf 96-01-01104}}
 \maketitle

\vspace{4mm}
\underline{\bf {I. Introduction.}}

\vspace{2mm}
The Gaussian curvature of a smooth surface embedded into the Euclidian
space is a
smooth function on the surface. We investigate below the local realization
problem :
given a germ of a smooth function of two variables, find a surface whose
Gaussian
curvature is the given function.

It is well known that any function germ $g(x,y)$ is realizable as the
Gaus\nolinebreak sian curvature of the surface $z = f(x,y)$, if the
curvature value
at the central point is not vanishing. It is also known, that it is
realizable (in
the same sense), if the curvature is vanishing at the central point, but its
differential does not vanish. In this case, the parabolic curve is smooth.

In the case of a singular parabolic curve, the problem is more difficult.
We shall
see, that any parabolic curve singularity occurs for a suitable surface :

\vspace{4mm}
\underline{\bf Theorem 1}. {\it For any smooth function of two variables
vanishing
at its critical point of finite multiplicity, there exists a smooth surface
in the
Eu\nolinebreak clidian $3$-space, whose Gaussian curvature coincides with
the given
function at a neighbourhood of the given point, (provided that the surface is
identified with the plane by a suitable local diffeomorphism, depending on the
function).}

I recall that the {\it multiplicity} $\mu$ of a critical point $0$ of
function $f$
is the dimension of the quotient vector-space of the space of formal power
series
in $(x,y)$ modulo the ideal generated by the partial derivatives of the
function :
$$\mu = {\rm dim}_\BbR \ \BbR \Bigl[ \lbrack x,y \rbrack \Bigr] \biggm/\!\!
\Bigl\{
a(x,y) f_x + b(x,y) f_y \Bigr\} \ . $$
Finite multiplicity critical points of holomorphic functions are just complex
isolated critical points. The codimension of the set of the functions
hav\nolinebreak ing a critical point of infinite multiplicity is infinite. Thus
typical finite-dimensional families of functions contain no function having a
critical point of infinite multiplicity.

\vspace{4mm}
\underline{\bf Corollary 1}. {\it Any singularity of finite multiplity
$\mu$ of a
plane curve is realizable as a singularity of the parabolic curve on a
surface in
the Euclidian $3$-space (up to a diffeomorphism).}

\vspace{4mm}
\underline{Remark 1}. The Theorem and the Corollary hold for analytical
functions
and diffeomorphisms as well as for the holomorphic ones or for the
infinitely smooth
ones and even for the finitely smooth ones (the number of the derivatives
in the last
case grows with $\mu$).

\vspace{4mm}
\underline{Remark 2}. I have no analytical (holomorphic, smooth)
counterexamples to
the statement of Theorem 1 for infinite multiplicity germs.

In the formal series setting, one can prove more :

\vspace{4mm}
\underline{\bf Theorem 2}. {\it For any formal power series $g (x,y) = ax +
by +...$
there exists a formal power series $f(x,y)$ such that the formal power
series of the
Gaussian curvature of the formal surface $z = f(x,y)$ coincides with $g$. }

In Theorem 2, the identification of the surface with the plane $(x,y)$ is
fixed by
the choice of the coordinates $(x,y,z)$ in the $3$-space.

\vspace{4mm}
\underline{\bf Corollary 2}. {\it Any formal singularity of a plane curve
$g(x,y) =
0$ is realizable as the singularity of the parabolic curve of some formal
surface $z
= f(x,y)$.}

\vspace{4mm}
\underline{Remark 3}. I have no counterexamples (even for $\mu = \infty)$
neither in
the analytical nor in the holomorphic nor in the smooth case (neither for
Theorem 2
nor for its Corollary), but I guess that the above series might diverge even for
finite $\mu$.

Theorem 1 follows from Theorem 2, as it is explained below (in paragraph
3). The same
arguments (in a simpler form) prove the following results.

\newpage
\underline{\bf Theorem 3}. {\it For any function $g(x,y)$ vanishing at a
critical
point of finite multiplicity, there exists a function $f(x,y)$, whose Hessien
$$h \ = \ h[f] \ = \ f_{xx} f_{yy} \ - \ (f_{xy})^2 $$
coindices with $g$ at a neighbourhood of the point up to a suitable smooth
change of
variables :
$$ g(x,y) \ = \ h \ \Bigl(X (x,y), \ Y(x,y) \Bigr) \ . $$
}

\vspace{4mm}
\underline{Remark 4}. Corollary 1 follows already from Theorem 3 (which is a
simplified version of Theorem 1) since the Hessien vanishes exactly along the
parabolic curve.

\vspace{4mm}
\underline{\bf Theorem 4}. {\it For any formal series $g(x,y) = ax +
by+...$, there
exists a formal series $f$ whose Hessien is $g = h[f]$.}

\vspace{4mm}
\underline{Remark 5}. Corollary 2 follows already from Theorem 4 (which is a
simplified version of Theorem 2).

\vspace{4mm}
\underline{Remark 6}. The proofs of these results provide in fact more :
one might
replace in Theorems 3 and 4 the Hessien $h = h[f]$ by any function (series)
of the
form, say,
$$g[f] \ = \ hM \Bigl( x,y,f_x, f_y, f,h \Bigr), \ M(0) \not = 0 $$
provided that $f(0) = f_x (0) = f_y(0) = 0$ (there exist many other
possibilities).

This Remark implies Theorems 1 and 2, since the Gaussian curvature admits such a
representation (in this case $M = M(f_x, f_y)$, see paragraph 4 below).

\vspace{4mm}
\underline{Remark 7}.  Theorems 1-4 have higher dimensional versions
(describing the
Hessiens of the functions of $n$ variables and the Jacobians of the Gaussian
mappings from the hypersurfaces in $\BbR^{n+1}$ to the $n$-sphere).

\vspace{4mm}
\underline{Remark 8}. Whether the statements of Theorems 3 and 4 hold for
infinite
mutiplicity critical points and for analytical (holomorphic, smooth)
functions $f$
and $g$ respectively is not known to me (see the Remarks 1 and 2 above).

\vspace{2mm}
The extension of the results of this paper to the (formally more general)
case of the
Jacobians of the Lagrangian mappings in symplectic geometry is automatical
(since
Lagrangian germs have gradient representatives).

\vspace{4mm}
\underline{\bf {2. Proof of Theorem 4.}}

\vspace{2mm}
\renewcommand{\theequation}{$\ast$}
\setcounter{equation}{0}
We shall look for the series
\begin{equation} f \ = \ f_2 + f_3 + ... \quad (f_i \ = \ {\rm terms \ of \
degree \ }
i) \ , \end{equation}
starting from $f_2 = y^2 \biggm/\!\! 2$. {\it This choice of $f_2$ is the
crucial point of the present paper} : the remaining part of the proof
consists of
standard calculations.

The terms of the Hessien $h[f] = h_1 + h_2 + ...$ are given by the obvious
formulae
$$h_1 = f_{3, xx}, \quad h_2 = f_{4,xx} + \Bigl( f_{3,xx} f_{3,yy} - f^2_{3,xy}
\Bigr) \ , $$
of the form
\renewcommand{\theequation}{$\ast \ast$}
\setcounter{equation}{0}
\begin{equation} h_k \ = \ f_{k+2, xx} + P_k \Bigl ( \lbrack f_m
\rbrack, \quad 3 \leq m < k +2 \Bigr) \ , \end{equation}
where $p_k$ is a (quadratic) polynomial with respect to the second derivatives
$\Bigl( f_{m,xx}$, $f_{m, xy}$, $f_{m, yy} \Bigr)$ of homogeneous functions
$f_m$.

The system of the equations for the functions $f_i$ that we get is triangular:
$$h_1 \Bigl[f_3 \Bigr] = g_1, \quad h_2 \Bigl[f_3, f_4 \Bigr] \ = \ g_2, ..., $$
$$h_k \Bigl[ f_3, ..., f_{k +2} \Bigr] \ = \ g_k, ... \ . $$

It remains to solve the equation
\renewcommand{\theequation}{$\ast \ast \ast$}
\setcounter{equation}{0}
\begin{equation} f_{k+2, xx} = g_k - P_k \Bigl( [f_3], ...,
[f_{k +1}] \Bigr) \end{equation}
(whose right hand side is known)
with respect to $f_{k +2} = \sum a_{p,q} x^p y^q$, $p+q = k +2$. Denoting
$g_k - P_k = \sum b_{p,q} x^py^q$, $p+q = k$, we solve the equation
$(\ast \ast \ast)$, taking
$$a_{p,q} = b_{p-2,q} \biggm/\!\! p(p-1) \quad {\rm if \ } \ p \geq 2 \ , $$
$a_{0,q} = a_{1, q} = 0$. Theorem 4 is proved.

\vspace{4mm}
\underline{Remark 9}. The extension of Theorem 4, mentioned in Remark 6, is
proved
by the same reasoning. Indeed, the multiplication of the Hessien by the
multiplier
$M = c + M_1+M_2 +..., c \not = 0$, preserves the triangular character of the system
$h[f] M[f,h] = g$ for $f$.

To study the $n$-dimensional hypersurfaces and functions of $n$ variables
(Remark
7) one should start from $f_2 = \pm y^2_1/2 \pm...\pm y^2_{n-1} /2$.

\vspace{4mm}
\underline{\bf {3. Proofs of Theorems 3 and 1}}

\vspace{2mm}
We deduce Theorem 3 from Theorem 4.

Let function $g$ has at the origin the critical value $0$ of finite multiplicity
$\mu$. For any $N$ the proof of Theorem 4 (in paragraph 2 above) provides a
polynomial $f$
such that $h[f] = g+r$, where the remainder $r$ has at the origin a zero of
order
at least $N$.

If $N \geq \mu +2$, the function $g+r$ is reducible to $g$ by a smooth
(analytical, holomorphic...) change of variables (see e.g., [1]). This proves
Theorem 3.

Theorem 1 follows similarly from Theorem 2 and from the finite $(\mu +2)$
determinacy
property of the finite multiplicity critical point (used in the proof of
Theorem 3
above).

\vspace{4mm}
\underline{\bf {4. Proof of Theorem 2.}}

\vspace{2mm}
We start from an obvious remark.

\vspace{4mm}
\underline{\bf Lemma}. {\it The Gaussian curvature $g$ of the surface,
defined in
some Carte\nolinebreak sian orthonormal coordinates $(x,y,z)$ in the
Euclidean space
as the graph of a function $z = f(x,y)$, is related to the Hessien $h$ of
the function
$f$ by the formula
$$g \ = \ h[f] \biggm/\!\! E^4 [f] \ , $$
where $E^2 \ = \ 1+ \Bigl( f_x)^2 + (f_y)^2 \ . $ }

\vspace{4mm}
\underline{Proof}. Function $E$  is the lenght of the normal vector $n = \Bigl(
-f_x, -f_y, 1)$ of the surface. The area element $dS$ of the surface in
inclined to
the horizontal direction at the same angle as the angle of deviation of the
normal
from the vertical direction. Hence $dS = E dx \wedge dy$.

The Gaussian mapping $G$ from the surface to the sphere is the product of
the gradent
mapping $F$ (sending $(x,y)$ to $(u = -f_x$, $v = -f_y$) from the
surface to the horizontal plane $\Bigl\{ u, v, 1 \Bigr\}$ (tangent to the
unite sphere at $(0,0,1)$), followed by the central projection $P$ from the
horizontal plane to the sphere: $G = P \circ F$.

The aera element of the sphere, $d\omega$, induces on the horizontal plane
the form
$P^\ast d\omega = E^{-3} du \wedge dv$ (the factor $E^{-2}$ is the
contribution of
the fact that the point of the plane is $E$-times farther from the origin
than the
point of the sphere, and an additional factor $E^{-1}$ is provided by the
inclination
of the horizontal plane to the radial direction).

By the definition of the Gaussian curvature we get
$$g \ = \ \frac{G^\ast d \omega}{dS} \ = \ \frac{J^\ast (E^{-3} du \wedge
dv)}{E dx
\wedge dy} \ = \ E^{-4} h \ , $$
since the Hessien of $f$ is the Jacobian of its gradient mapping $F$ :
$F^\ast du
\wedge dv $ = $ h\lbrack f \rbrack dx \wedge dy$. The Lemma is thus proved.

To prove Theorem 2, we look for the series $(\ast)$ as in paragraph 2. We get
$$E^{-4} \lbrack f \rbrack = 1 + e_2 \lbrack f_2 \rbrack + e_3 \Bigl(
\lbrack f_2
\rbrack, \lbrack f_3 \rbrack \Bigr) + ..., $$
$e_k$ being the homogeneous part of degree $k$ of the power series $E^{-4}$ in
$(x,y)$.

The form $e_k$ is a weighted homogeneous polynomial in the first
derivatives of the
functions $f_2, ..., f_k$ (higher order terms, starting from $f_{k+1}$,
contribute
noting to $e_k$) :
$$e_2 \lbrack f_2 \rbrack = 2y^2, \quad e_3 \Bigl( \lbrack f_2 \rbrack,
\lbrack f_3
\rbrack \Bigr) \ = \ - 4y f_{3,y}, \ ... \ , $$
$$e_k \ = \ e_k \Bigl( \lbrack f_2 \rbrack, ..., \lbrack f_k \rbrack \Bigr)
\ . $$

It follows that the system
$$h\lbrack f \rbrack \ E^{-4} \ \lbrack f \rbrack \ = \ g $$
of the equations for the unknowns $f_k$, (which we should solve to prove
Theorem 2,
conformally to the Lemma) is triangular.

Indeed, equating the homogeneous terms in the product
$$ \Bigl( h_1 + h_2 + ... \Bigr) \ \Bigl( 1+e_2 + ... \Bigr) \ = \ g_1 +
g_2 + ... ,
$$
we get for the unknowns $f_k$ the sequence of equations
$$h_1 \ = \ g_1, \quad h_2 = g_2, \quad h_3 + h_1 e_2 = g_3, ..., $$
$$h_k + Q_k \ \Bigl( \Bigl[ h_{<k} \Bigr], \Bigl[ e_{<k} \Bigr] \Bigr) \ =
\ g_k \ .
$$
Substituing the expression $(\ast \ast)$ from the proof of Theorem 4 to $h_k$, we
get for the unknown functions $f_k$ the system of equations of the form
$$f_{k+2, xx} \ = \ - R_k \Bigl( \Bigl[ f_{<k+2} \Bigr] \Bigr) + g_k \quad
(k \geq
1) $$
(with known $R_k$ and $g_k$). We solve these equations the same way as the
equations
($\ast \ast \ast$) were solved in paragraph 2, finding successively $f_3,
f_4, ...$.
Theorem 2 is thus proved.

As it was explained in paragraph 3, Theorem 1 follows.

\vspace{4mm}
\underline{\bf {5. Parabolic curves at a flattening point}}.

\vspace{2mm}
We call {\it the flattening points} of the surface $z = f(x,y)$ the points
where the
second differential of $f$ vanishes. In suitable (cartesian orthonormal)
coordinates
the Taylor series of $f$  at a flattening point takes the form
$$f \ = \ f_3 + f_4 +.... \quad \Bigl( f_0  =  f_1  =  f_2  =  0 \Bigr) \ . $$

A typical example is the generalized monkey saddle $D^\pm_4$ : $f = 3x^2 y
\pm y^3$. The flattening points are critical points of the Gaussian
curvature, where
the critical value vanishes.

\vspace{4mm}
\underline{\bf Theorem 5}. {\it The second differential of the Gaussian
curature at
a flattening point can't be a nonzero nonnegative quadratic form (in
particular, the
flattening point can't be a nondegenerate minimum point of the Gaussian
curvature),
while all the other quadratic forms are realizable as the second
differentials of
the Gaussian curvature at the flattening points.}

\vspace{4mm}
\underline{Proof}.  Let $f_3 = ax^3 + 3bx^2 y + 3cxy^2 + dy^3$. The Hessien
of this
function is
$$h_2 \ = \ 36 (A x^2 + Bxy + C y^2) \ , $$
where $A = ac - b^2, \ B = ad-bc, \ C = bd - c^2$.

Choose the (cartesian orthonormal) coordinates reducing the quadratic form
$h_2$ to
the normal form $(B = 0)$. In these coordinates $ac = A+b^2$, $bd = C+c^2$,
$ad =
bc$. Hence we have
$$(A+b^2) (C+c^2) \ = \ b^2c^2, \quad AC + Ac^2 + Cb^2 \ = \ 0 \ . $$
If the form $Ax^2 + Cy^2$ is positive definite $(A> 0, \ C > 0)$ the last
equality
is impossible. The positive semidefinite case ($A> 0, C = 0$) is impossible
too, since
then $c = 0$ and hence $A+b^2 = 0$.

If one of the coefficients, say $A$, is negative, the required realization is
provided by
$$b \ = \ \pm \sqrt{-A} \ , \ a = c = 0, \ d = C \bigm/\!\! b \ . $$
Theorem 5 is thus proved.

\vspace{4mm}
\underline{Remark 10}.  Our formulae imply that the degenerate forms $h_2$ are
realized only by degenerate forms $f_3 = ax^3 + 3bx^2y$, and that the zero-form
$h_2$ is realized only by the bidegenerate forms $f_3 = ax^3$.

\vspace{4mm}
\underline{\bf Theorem 6}. {\it The parabolic curve singularity at a flattening
point of a surface can't be diffeomorphic to the $E_6$ singularity (to the
singularity of the curve $x^3 + y^4$ at the origin). }

\vspace{4mm}
\underline{Proof}. The equation of the parabolic curve of the surface $z =
f(x,y)$
is $h\lbrack f \rbrack (x,y) = 0$. Choose a coordinate system for which $f
= f_3 +
f_4 + ...$ . We should have $h_2 = 0$ to realize the $E_6$ singularity of
$h$. From
the Remark 10 above, we get (in suitable coordinates) $f_3 = ax^3$. We can
normalize
$a$ to be $1/6$ ($a \not = 0$, since for $E_6$ $h_3 \sim x^3 \not = 0$).

Now compute $h_3$. Since we should get the $E_6$ singularity for $h \lbrack f
\rbrack$, it has the form $h_3 = (px + qy)^3$. But for $f_2 = 0$, $f_3 = x^3
\bigm/\!\! 6$ we get $h_3 = xf_{4,yy}$. Hence $q = 0$, $h_3 = Px^3$, $P
\not = 0$.
Now we get $f_{4, yy} = Px^2$, $f_4 = x^2 e_2(x,y)$,
$$h_4 \ = \ xf_{5,yy} + f_{4,xx} \ f_{4,yy} - (f_{4,xy})^2 \ . $$
Since $f_4$ is divisible by $x^2$, $f_{4,yy}$ and $f_{4,xy}$ are divisible
by $x$,
and thus $h_4$ is divisible by $x$.

Hence the sum of the terms of degrees $3$ and $4$ of the Taylor series of
$h\lbrack
f \rbrack$ is equal to
$$ H(x,y)  =  h_3 + h_4  =  Px^3 + x R_3 (x,y) \ . $$
We shall prove now that such a function can't have an $E_6$ singularity (whose
$4$-th degree Taylor polynomial should be equal $X^3 + Y^4$ for some local
coordinates $(X,Y)$, defined by $(x = \alpha X + \beta Y + ...,$ $y = \gamma X +
\delta Y = ...)$).

Denote the expression of $H$ in new coordinates by $K (X,Y) = $ \break \hfill
$ H(x(X,Y),  y
(X,Y))$. Equating the terms of degree 3 to $X^3$, we get
$$P (\alpha X + \beta y)^3 \ = \ X^3 \ . $$

Thus $\beta = 0$, $x = \alpha X$ + (higher order terms).

It follows that all the terms of degree $4$ in the series $K$ are divisible
by $X$.
Indeed, the terms of degree $4$, coming from $P x^3$, have the form
$$T_4 = \Bigl( 3P x^2 (x -\alpha X) \Bigr)_4 \ = \ 3P (x^2)_2 (x - \alpha
X)_2 \ . $$
The quadratic part of the expansion of $x^2$ in the power series with
argument $X,Y$
is equal to $\alpha^2 X^2$ since $\beta = 0$. Hence the polynomial $T_4$ is
divisible by $X^2$.

The fourth degree terms coming from $xR_3$ do contain $X$ for the same
reason $(x =
\alpha X$ + higher order terms). Hence the $4$-th degree Taylor polynomial
of $K$ is
divisible by $X$. Thus it can't be equal to $X^3 + Y^4$. Theorem 6 is proved.

\vspace{4mm}
\underline{Remark 11}. I do not know which curve singularities are realizable as
the parabolic curve singularity at a flattening point. This question
(together with
similar questions on the singularities of the Hessien of the Gaussian
curvature and
of the Jacobian of a Lagrangian mapping) might be interesting also for higher
flattenings.

\vspace{4mm}
\begin{thebibliography}{1}

\item  V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, {\it Singularities of
differentiable maps}, \underline {Vol. 1}, Nauka, Moscow, 1982 (English
translation :
Birkh\"{a}user, 1985).

\end{thebibliography}


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\hfill 75775 PARIS CEDEX 16 (FRANCE)

\smallskip
\hfill and

\smallskip

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\hfill 117966 MOSCOW,  GSP-1 , (RUSSIA)

\end{document}
