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\topmatter
\title
Remarks on the Parabolic Curves on Surfaces
and on the Many-Dimensional M\"obius--Sturm Theory
\endtitle
\thanks
Supported by the Russian Foundation for Basic Research,
project 96-01-01104.
\endthanks
\author V. I. Arnold \endauthor
\affil V.~A.~Steklov Mathematical Institute \endaffil
\date August 26, 1997 \enddate
\endtopmatter

\head Conjecture on Four Parabolic Curves\endhead

According to the classical M\"obius theorem, a curve on the
projective plane has at least three
inflection points provided that this curve is close to a projective line.
This theorem is a special case of the
Sturm--Hurwitz theorem \cite2 stating that the number of the
sign alternations of a function on a circle is not less than the
number of the sign alternation of the lowest harmonic entering its
Fourier series with nonzero coefficient. The M\"obius theorem can be
extended to all noncontractible curves embedded in the projective
plane. The Sturm theorem has probably a global version too.
Attempts to extend the M\"obius and Sturm theorems to the
many-dimensional case (i.e., to surfaces in the projective space and
to functions of several variables) show that natural analogs of
inflection points are parabolic curves of surfaces and zeros of
the Hessians of functions. This means that the differential operators
corresponding to surfaces and to functions of two variables are
quadratic (of degree $n$ in the case of
$n$-dimensional manifolds and functions of $n$ variables)
rather than linear (as is the case in the ordinary
Sturm theory of functions of one variable).
According to the general principle of topological economy of
algebro-geometric objects, one might expect that the minimal number
of parabolic curves (say, on surfaces which are small
deformations of a projective plane in the projective space) is
attained in the case of cubic surfaces. For such cubic surfaces, the
number of parabolic curves (calculated by B. Segre \cite1) is four.
This leads to the {\it Aicardi conjecture\/} (see \cite3): {\it the
number of parabolic curves on a generic surface in the projective
space is at least four provided that the surface is sufficiently
close to a projective plane\/ \rom(in the smooth topology\/}).
One might even conjecture (by analogy with the M\"obius theorem and
with the theory of Lagrangian intersections and Legendre linkings in
symplectic and contact topology) that the number of parabolic
curves is still not less than four for large deformations of the
projective plane as well (at least while the deformed surface remains
embedded) and that the corresponding results of the
``set-valued Morse theory for higher derivatives'' hold for more general
problems of symplectic and contact topology with nonholonomic
constraints. However, at present even the simplest special case of
these conjectures stated above is neither proved nor disproved. This
special case will be studied below.
Our strategy is as follows. We choose a construction of perturbations
of the projective plane for which we can inspect the number of
parabolic curves. We try to choose the construction parameters so
that the deformed surface would have less than four parabolic curves.
If such a choice is possible, then we find a counterexample to the
conjecture. If, for any choice of the parameters, we obtain at least
four parabolic curves, then we receive a confirmation of the
conjecture and some (possibly nontrivial) information on the
deformations of the class under consideration (for instance, on
the ``spherical functions with singularities'' introduced below).
Possibly the most natural approach is to consider the polyhedral
deformations constructed from the triangulations of the perturbed
projective plane. On deforming the resulting polyhedron into a generic
smooth surface by some specially chosen construction we might probably
describe the number of occurring parabolic curves on the surface in
terms of the combinatorics of the polyhedron (taking into account the
singular curvatures concentrated at edges and vertices). In this way
one might reformulate topological conjectures on parabolic curves as
combinatorial statements on the triangulations of the projective
plane.
However, in the present paper, such a smoothing is not introduced
and, instead of a combinatorial perturbation, we consider another
perturbation of the projective plane. This perturbation is hyperbolic
almost everywhere except for finitely many points including the
logarithmic poles (in a neighborhood of each of this poles, the
perturbation can be smoothed on creating a small parabolic curve
and an elliptic component).
These ``hyperbolic'' perturbations of the projective plane are
constructed by means of the singular odd spherical functions $u$
(that satisfy the equation $\De_Su+2u=g$ on $S^2$, where $\De_S$ is
the spherical Laplacian and $g$ is a linear combination of delta
functions). We shall prove that {\it the number of parabolic curves
on a deformed surface generated by this construction is at least
four.}
Another construction of perturbations of the projective plane
involves the decomposition of the projective plane into a (large)
disk and a (thin) M\"obius band, and a perturbation is performed on
one of this two domains. For instance, one can treat a perturbation
defined by a cubic surface over the M\"obius band. Now, to study the
perturbations over the disk, it suffices to study the zeros of the
Hessian of the function that defines the perturbation in an affine
coordinate system and satisfies given boundary conditions (provided
by the chosen perturbation outside the disk).
Thus, we arrive at the problem on the plane parabolic curves defined
by the equation $F_{xx}F_{yy}=F^2_{xy}$.
The study of these curves is equivalent to the study of the critical
points and caustics of the gradient mappings. Our results imply, for
instance, lower estimates for the number of cusps on the caustics of
a gradient mapping of the disk satisfying given boundary conditions.
For example, {\it if a generic potential coincides with the function
$F=x^3-3xy^2$ outside some large disk, then the gradient mapping
has at least three $A_3$ singularities inside this disk\/ \rom(which means
that the caustic has at least three cusps\/}).
This example shows that the gradient mappings are distinguished by
their topological properties in the larger class of Lagrangian
mappings (in which our mapping can be deformed into a mapping with
only one cusp, according to Entov and Eliashberg).
Another way to inspect parabolic curves is to start from a
surface with a degenerate parabolic curve and to deform it into a generic
surface (in a neighborhood of the curve). For instance, for the
surface $z=1/(x^2+y^2)$, the line at infinity is a degenerate
parabolic curve.
In this case and in several similar examples we show that, {\it under
a generic one-parameter small deformation of the surface, a
degenerate parabolic curve decomposes into nondegenerate parabolic
curves whose total number is at least four}.
In the above example, the line at infinity gives birth to at least
three parabolic components (actually, by the M\"obius--Sturm theorem).
I conjecture that, in fact, the results that are proved below only
for small one-parameter deformations are valid in an entire
neighborhood of our surface in the space of deformed surfaces (not
only in our special examples but also for all surfaces with
degenerate (multiple) parabolic curves). Should the Aicardi conjecture be
true, it would prove these results even for very large deformations
of the surface provided that this surface remains fixed outside a
neighborhood of the degenerate parabolic curve.
\ah\bS2.\enspace Equations of Parabolic Curves\endah
We describe the projective space $\Bbb RP^3$ by means of the
homogeneous coordinates $[x\>{:}\>y\>{:}\>z\>{:}\>w]$. We start from
the unperturbed plane $\Bbb RP^2$ defined by the equation $w=0$. Any
perturbed surface is defined by an equation of the form %%$w=f(x,y,z)$,
where $f$ is a homogeneous function of degree one (and thus an odd
function). Let us find the parabolic curves on the perturbed surface.
The following result is classical.
\tm{Lemma 1} The parabolic points of the surface $w=f$ are
represented on the sphere $x^2+y^2+z^2=1$ by the critical points of
the restriction of the gradient of the function $f$ to this sphere.
\endtm
\pf{Proof} Near a parabolic point, a surface is exceptionally well
approximated by a plane. A plane in $\Bbb RP^3$ is defined by a
linear function $f$. The gradient of linear function is constant. If
$f$ is exceptionally well approximated by a linear function at a
given point, its gradient is exceptionally well approximated by a
constant. This means that this point is critical for the gradient of
$f$.\qed
\endpf
Here one can replace the sphere (in a neighborhood of the point under
consideration) by any surface transversal to the radial rays (since
the gradient of a homogeneous function of degree one is constant
along each of these rays). For instance, one may take the plane
tangent to the sphere at this point.
\ex{Example} The domain $z\ne0$ is served by the plane $z=1$. Denote
the restriction of $f$ to this plane by $F(x,y)=f(x,y,1)$. We may
consider the relation $w=F(x,y)$ as the affine equation of the
perturbed surface in the affine chart $z=1$ of the projective space.
We obtain the following obvious formulas.
\endex
\tm{Lemma 2} In our affine coordinates, the equation of the curve of
parabolic points of the surface $w=f$ acquires the form
$$
\op{Hess}F=\op{det}\pmatrix F_{xx}&F_{xy}\\
F_{xy}&F_{yy}\endpmatrix=0. \tag1
$$
\endtm
\pf{Proof} For $z=1$, the equation of the tangent plane $w=ax+by+cz$
becomes $w=ax+by+c$. Hence, the coefficients $a$, $b$, and $c$
are equal to the values of $F_x$, $F_y$, and $F$,
respectively, at our point.
The deviation of the surface from its tangent plane is measured (in
the quadratic approximation) by the second differential of $F$ at
the tangency point. This quadratic form is degenerate if and only if
$\op{Hess}F=0$.\qed
\endpf
Introduce the {\it polar coordinates\/} (\<$x=R\cos\ph$,
$y=R\sin\ph$) on the plane $z=1$. We shall use the archaic notation
for the derivatives: $F_R=(\pa/\pa R)F(R\cos\ph,R\sin\ph$), and
so on.
\tm{Lemma 3} In the polar coordinates, the equation for the curve of
parabolic points of the surface $w=f$ reads
$$
\op{det}\pmatrix F_{\ph\ph}+RF_R&F_{\ph R}-F_{\ph}/R\\
F_{\ph R}-F_{\ph}/R&F_{RR}\endpmatrix=0.\tag2
$$
\endtm
\pf{Proof} Choose a point $(R_0,\ph_0)$. The tangent plane at this
point has the equation $w=aR\cos\ph+bR\sin\ph+c$. By equating the
$1$-jet of $F$ with the $1$-jet of the right-hand side
of this equation at our point, we obtain the
following equations for $(a,b,c)$:
$$
\align
F^{(0)}&=aR_0\cos\ph_0+bR_0\sin\ph_0+c,\\
F_R^{(0)}&=a\cos\ph_0+b\sin\ph_0,\\
F_{\ph}^{(0)}&=-aR_0\sin\ph_0+bR_0\cos\ph_0
\endalign
$$
(the superscript $0$ stands for the evaluation at our point). The
second derivatives of the difference $F-aR\cos\ph-bR\sin\ph-c$ at
our point form the matrix on the left-hand side in~(2). Its
determinant vanishes just at the parabolic points.
\endpf
To study the surface in a neighborhood of the point at infinity
(\<$z=0$), it is convenient to use the {\it cylindrical
coordinates\/} (\<$r\=1/R$, $\ph$). This chart of the projective
plane is defined by the central projection of the sphere
$x^2+y^2+z^2=1$ on the cylinder (\<$x=\cos\ph$, $y=\sin\ph$, $z=r$)
that is tangent to the sphere along the equator. The opposite points
of the cylinder, $(r,\ph)$ and $(-r,\ph+\pi)$, represent the same
point of the projective plane. The parabolic curve is cooriented, and
hence each of its components is represented on the cylinder by two
ovals.
A point (\<$x=R\cos\ph$, $y=R\sin\ph$, $z=1$) of the plane $z=1$
corresponds to the point (\<$x=\cos\ph$, $y=\sin\ph$, $z=r=1/R$) of
the cylinder. On this cylinder, the homogeneous equation of the
perturbed surface $w=f(x,y,z)$ becomes $w=R^{-1}F(R,\ph)$
(certainly, we mean the expression of the function $F$\< defined on
the plane via the polar coordinates).
Let us rewrite the last equation in the form $w=H(r,\ph)$. The functions $H$
and $F$ are related as follows:
$$
H(r,\ph)=rF(1/r,\ph),\qquad F(R,\ph)=RH(1/R,\ph).\tag3
$$
The function $H$ is smooth at $r=0$. It satisfies the relation
$$
H(-r,\ph+\pi)=-H(r,\ph)
$$
because the function $w=f(x,y,z)$ is odd. Hence, the Taylor
coefficients of the expansions
$$
\align
H(r,\ph)&=h_0(\ph)+rh_1(\ph)+\topsmash{\frac{r^2}2}\,h_2(\ph)+\dots,\\
F(R,\ph)&=Rh_0(\ph)+h_1(\ph)+\frac1{2R}\,h_2(\ph)+\dots
\endalign
$$
satisfy the oddness conditions
$$
h_{2k}(\ph+\pi)=-h_{2k}(\ph),\qquad h_{2k+1}(\ph+\pi)=h_{2k+1}(\ph).\tag4
$$
\tm{Lemma 4} In the cylindrical coordinates, the equation of the
parabolic curve of the surface $w=f$ has the following form at
infinity:
$$
\op{det}\pmatrix H_{\ph\ph}+H-rH_r&H_{r\ph}\\
H_{r\ph}&H_{rr}\endpmatrix=0.\tag5
$$
\endtm
\pf{Proof} Arguing as in the proof of Lemma 3, we obtain
$w=a\cos\ph+b\sin\ph+cr$. Hence, $(a,b,c)$ satisfies the equations
$$
H^{(0)}=a\cos\ph_0+b\sin\ph_0+cr_0,\quad H_r^{(0)}=c,\quad
H_{\ph}^{(0)}=-a\sin\ph_0+b\cos\ph_0.
$$
The second derivatives of the difference $H-a\cos\ph-b\sin\ph-cr$
form the matrix in Eq.~(5).\qed
\endpf
We also need the following assertion.
\tm{Lemma 5} Suppose that a homogeneous function $f$ of degree one is
harmonic. Then the corresponding quadratic form that measures the
deviation of the surface $w=f$ from its tangent plane is either
hyperbolic or zero\/ \rom(thus, the above three Hessians \rom{(1)},
\rom{(2)}, \rom{(5)} are either negative or zero\/\rom; in the latter
case, all entries of the corresponding matrix vanish\/\rom).
\endtm
\pf{Proof} Without loss of generality we can assume that our point
has the coordinates (\<$x=y=0$, $z=1$). At this point we have
$f_{zz}=0$ because $f$ is homogeneous of degree one; however,
$f_{xx}=F_{xx}$ and $f_{yy}=F_{yy}$. Hence, the trace
$F_{xx}+F_{yy}$ of the Hessian matrix is zero. Thus, at our point
we have $\op{Hess}F=-F_{xx}^2-F_{xy}^2\le0$.\qed
\endpf
\rk{Remark} The restriction of a first-order
homogeneous harmonic function in the space
to the unit sphere is an eigenfunction
of the spherical Laplacian with eigenvalue $-2$, i.e.,
$\De_Su+2u=0$. The spherical functions of degree one are simply the
linear functions $ax+by+cz$. However, we can also apply Lemma 5 to
``singular spherical functions'' satisfying $\De_Su+2u=g$, where $g$
is an odd distribution on the sphere with support consisting of
finitely many points.
The solutions of this equation define singular perturbations
of the projective plane related to surfaces whose second quadratic
form is hyperbolic everywhere except for the poles at the points of
the support of $g$ and the points of total degeneration (at which the
entire form is zero).
One can use these special singular perturbations to construct regular
perturbations for which we can inspect their parabolic curves (with a
desired curve near every logarithmic pole of $u$ and near each point
of total degeneration provided that $g$ is a generic combination of
delta functions).
\endrk
\de{Definition} A function on a sphere is said to be {\it hyperbolic\/}
if the corresponding quadratic form is hyperbolic everywhere except
for finitely many points (where the function may be singular).
The simplest singularity of a hyperbolic function is a (logarithmic)
pole.
\endde
\tm{Conjecture} An odd hyperbolic function on the $2$-sphere all of
whose singularities are the simplest ones has at least six
logarithmic poles.
\endtm
\ah\bS3.\enspace Singular Spherical Functions\endah
By a {\it singular spherical function\/} we mean a function
satisfying the equation of spherical functions everywhere except for
finitely many singular points (at which the result of application of
the corresponding operator to the function is a finite-order
distribution, that is, a linear combination of delta functions
and their derivatives).
\tm{Lemma 6} An odd singular spherical function of degree one on the
$2$-sphere all of whose singularities are the simplest\/
\rom(logarithmic\/\rom) poles has at least six such poles. If it has
exactly six poles, then they all belong to the same great circle of
the sphere.
\endtm
\pf{Proof} By assumption, the function $\De_Su+2u=g$ is a linear
combination of finitely many delta functions with positive
coefficients at $k$ points and with opposite coefficients at the
opposite points of the sphere. By the Fredholm alternative, the above
equation for $u$ is solvable if and only if $g$ is orthogonal to the
solutions of the equation $\De_Su+2u=0$, that is, to the linear
functions. This orthogonality condition means that the mass center of
the $k$ positive charges of the distribution $g$ coincides with the
center of the sphere. The mass center of two points on the sphere
coincides with the center only if the points are opposite and the
masses are equal. In this case $g=0$. Hence, $k\ge3$. The mass center
of three points on the sphere coincides with the center only if the
plane containing these points passes through the center. In this case,
the three opposite points belong to the same plane.\qed
\endpf
\tm{Theorem 1} A generic surface obtained from the projective plane
by a deformation generated by a generic odd singular spherical
function with the simplest poles\/ \rom(by smoothing
and by a small generic deformation of this function\/\rom) has at least
four parabolic curves.
\endtm
\pf{Proof} Near each of the poles, smoothing creates a small
parabolic curve bounding an elliptic domain.
The total vanishing of the form is impossible for the generic
surfaces, but it can stably occur at individual points on hyperbolic
surfaces. One can construct a hyperbolic surface by using spherical
functions or solutions of other elliptic differential equations. For
instance, for the usual harmonic functions $u=\op{Re}U$ on the plane
of the variable $z=x+iy$, the three second partial derivatives of $u$
vanish at the zeros of the second derivative of the holomorphic
function $U$\<, and these zeros persist under deformations of the
holomorphic function.
According to Lemma 6, the number of poles (and hence the number of
parabolic curves generated by the poles under smoothing) is at
least six. Besides these singularities, points of the total
degeneration of the second differential must also exist. Let us prove
this fact.
{\it There is a field of asymptotic directions in the hyperbolic
domain of the sphere\/ \rom(that is, in the complement of the poles
and the points of total degeneration \/}). Indeed, a field of
hyperbolic quadratic forms on the tangent planes of the sphere is
well defined in the hyperbolic domain. A straight line bounding from
the right the sector in which the form is positive is defined by the
orientation of the sphere unambiguously. These (nonoriented) lines
form the desired field of asymptotic directions.
The sum of the indices of the singular points of this field is $2$ (the
indices can be half-integers because the lines of the field are not
oriented). But the $2k$ poles contribute $2k$ to the index (since the
index of the field of asymptotic lines is equal to one at every
simplest pole (and possibly at any point at which the hyperbolic
function tends to $\infty$)). Hence, the sum of indices of the
points of total degeneration is $2-2k\le-4$. In particular, the index
of at least one of these points is nonzero.
We shall prove that, under a generic deformation of a hyperbolic
surface (or function), such a point generates a small parabolic
curve. Indeed, consider the ``second Gauss'' mapping of the plane
$(x,y)$ into the three-dimensional space of the quadratic forms
$A\,dx^2+2B\,dx\,dy+C\,dy^2$ given by the formulas
$A=F_{xx}$, $B=F_{xy}$, and $C=F_{yy}$ (locally, in a
neighborhood of our point of total degeneration, which we can assume
to have the coordinates (\<$x=0$, $y=0$, $z=1$)). The degenerate
forms constitute the cone $AC=B^2$\<. The parabolic points are the
points of the preimage of this cone under the second Gauss mapping.
Since $F$ is hyperbolic, it follows that a punctured neighborhood
of the point $x=y=0$ at which the form is totally degenerate is sent
by the second Gauss mapping into the domain $AC<B^2$ of hyperbolic
forms. This domain is the nonsimply connected exterior component of
the complement of the cone.
{\it The index of our isolated singular point of the field of
asymptotic directions is proportional\/ \rom(with a nonzero
coefficient\/\rom) to the linking number of the image of the boundary
circle of our neighborhood with the axis of the cone}. Hence, if the
index is nonzero, then the linking coefficient is nonzero. After a
small generic deformation of $F$ (under which $F$ becomes
nonhyperbolic), the image of neighborhood no longer contains the
vertex of the cone $(A=B=C=0)$, but this image intersects the cone
along a closed curve. Hence, our point of total degeneration gives
birth to a small parabolic curve under a generic deformation of the
initial hyperbolic function $F$.
Together with the three parabolic curves born near the three poles,
we obtain at least four parabolic curves on $\Bbb RP^2$.\qed
\endpf
\rk{Remark} Our reasoning gives $3k-2$ parabolic curves on $\Bbb
RP^2$ provided that the initial odd hyperbolic function on the sphere
has $2k$ simplest poles and only the simplest points of total
degeneration (which possibly holds for odd singular spherical
functions under a generic distribution of the charges of $g$).
Indeed, the simplest points of total degeneration are the monkey
saddles similar to that defined by the harmonic function
$F=\op{Re}\(x+iy)^3$ at the origin. In this simplest case, the index
of the field of asymptotic directions is equal to $-1/2$, and the
coefficient of linking with the axis of the cone is equal to $-1$ (if we
choose the natural orientation on the plane
$A+C=0$ of zero-trace forms).
In this case, the second Gauss mapping of a neighborhood of
the point of total degeneration into the hyperbolic domain of the
space of forms $(A,B,C)$ is transversal to the axis of the cone at
the point of total degeneration $x=y=0$.
\endrk
Hence, on the sphere we obtain $N$ simplest points of total
degeneration with the sum of the indices $2-2k=-N/2$. Hence, the
number $N/2$ of the simplest points of total degeneration on $\Bbb
RP^2$ is $2k-2$. Together with $k$ poles, this gives $3k-2$ parabolic
curves. If the points of total degeneration are not the simplest
ones, then we seemingly can obtain only four parabolic curves, as can
be shown by example in which equal positive charges form a regular
triangle.
One may obtain singular spherical functions explicitly from
homogeneous rational harmonic functions in $\Bbb R^3$\<. On this way
one can find the simplest odd singular spherical function of degree
one whose only singularities are the poles of the sphere (the
calculations are very simple in the coordinates $\xi=x+iy$,
$\eta=x-iy$). These singularities are multipole: in our notation,
$$
F=(2R^{-2}+3)\cos2\ph.
$$
Under a perturbation of the singular function $F$, one obtains at
least four parabolic curves on $\Bbb RP^2$: two are born from the two
(simplest) points of total degeneration at infinity and the other two
at the maximum and at the minimum of the smoothed singularity at
the origin.
\tm{Conjecture} The number of parabolic curves generated by
smoothing an odd hyperbolic function on the $2$-sphere and under
a small deformation making it generic, is at least four, for
arbitrary finite-order poles.
\endtm
This is not proved even for the rational spherical functions of
degree one (but this certainly follows from the Aicardi conjecture).
\ah\bS4.\enspace Deformations of a Degenerate Parabolic Curve\endah
Consider the cubic surface defined in the affine chart $z=1$ by the
equation $w=1/(x^2+y^2)$ (we shall later smooth the singularity at
the origin). This cubic surface, for which $F=R^{-2}$ and $H=r^3$, is
hyperbolic in its regularity domain over the $(x,y)$-plane and has a
degenerate parabolic curve at infinity (\<$r=0$). Perturbations
of this surface near infinity are described in the cylindrical
coordinates by the function $H=r^3+\e h(r,\ph)$, where $\e\ll1$ and the
perturbing function $h$ satisfies the oddness condition
$h(-r,\ph+\pi)=-h(r,\ph)$.
\tm{Theorem 2} For a sufficiently small $\e$, the degenerate
parabolic curve at infinity of the original surface transforms into
at least three nondegenerate parabolic curves of the perturbed
surface, provided that the perturbing function $h$ is generic.
\endtm
\pf{Proof} The equation of parabolic points (5) has the form
$$
-12r^4+\e\(6r\(h_{\ph\ph}+h-rh_r)-2r^3h_{rr})+O(\e^2)=0.\tag6
$$
Let us introduce a new variable $\rho$ by the scaling $r=\e^{1/3}\rho$.
Equation (6) becomes
$$
\e^{4/3}[\rho\(-12\rho^3+6a(\ph))+O(\e^{1/3})]=0,\tag7
$$
where $a=(h_{\ph\ph}+h)|_{r=0}$. First we neglect the term with $O$
and obtain
from (7) a curve consisting of two smooth branches, $\rho=0$ and
$\rho=(a(\ph)/2)^{1/3}$, which intersect each other transversally at
the zeros of the function $a$ (which are simple for generic $h$).
According to the Sturm theorem, {\it the number of these zeros is at
least six}. Indeed, by (4), for $r=0$ we have $h(\ph+\pi)=-h(\ph)$.
Hence, all harmonics of $h$ are of odd order. It follows that the
lowest harmonic of $a$ is of order not less than three and has at
least $6$ zeros.
The product $\rho(-12\rho^3+6a(\ph))$, where the zeros of $a$ are
simple, is a Morse function in neighborhoods of the saddle points
placed at the $2k$ zeros of the function $a$. A perturbation of the
product defined by the $O$-term in (7) moves these $2k$
saddles to nonzero levels, and thus the pair of intersecting branches
at the zero level of the function (7) transforms into a nonsingular
curve. The Hessian nature of Eq.~(7) determines the signs of all $2k$
surgeries.
\endpf
\tm{Lemma 7} The smooth curve\/ \rom{(7)} defined by a generic
perturbation $h$ consists of $2k$ components that bound the domains
close to the positivity domains of the product
$\rho\(-12\rho^3+6a(\ph))$.
\endtm
\pf{Proof} Consider a plane in the space of quadratic forms
$A\xi^2+2B\xi\eta+C\eta^2$ that intersects the cone of degenerate
forms $AC=B^2$ along two lines. Under a small generic deformation of
this plane we obtain a plane whose intersection with the cone forms a
hyperbola, and the discriminant $AC-B^2$ is negative between the
two branches of the hyperbola. In other words, from {\it the four
quadrants of the nonperturbed plane that consist of elliptic and of
hyperbolic forms alternately, the two opposite hyperbolic quadrants
merge under a generic deformation, while the elliptic part
becomes separated by the hyperbolicity domain on the perturbed plane.}
Hence, the number of the components born at infinity is $2k\ge6$ on
the cylinder, that is, $k\ge3$ on~$\R P^2$\<. Theorem 2 is thereby
proved.\qed
\endpf
It follows from the theorem that smoothing our surface at the
origin followed by a deformation making it generic at infinity
makes it a smooth surface with at least four parabolic curves.
\rk{Remark} It seems that {\it the number of parabolic curves on any
surface defined by a function of the form $H=r^3+h(r,\ph)$,
where $h(-r,\ph+\pi)=-h(r,\ph)$, is at least three provided that the
perturbation $h$ is generic and sufficiently small \rom(and this
possibly holds for any generic $h$ vanishing for $|r|>1$\<\rom)}. It
is proved above only for one-parameter families of deformations.
\endrk
Now consider the cubic surface defined by the equation
$w=x/(x^2+y^2)=R^{-1}\cos\ph$ (in the affine chart $z=1$). The
function is harmonic and hence hyperbolic (and singular at the
origin). The surface has a degenerate parabolic curve at infinity
(where the surface is defined by the function $H=r^2\cos\ph$ in the
cylindrical coordinates of \S2). Equation~(5) of parabolic points
becomes $-4r^2=0$. Near the line at infinity the perturbed surface is
defined by the function
$$
H=r^2\cos\ph+\e h(r,\ph),\qquad\e\ll1,\quad h(-r,\ph+\pi)=-h(r,\ph).
$$
\tm{Theorem 3} For sufficiently small values of the perturbation
parameter $\e$, the degenerate parabolic curve at infinity is
transformed into at least two nondegenerate parabolic curves of the
corresponding perturbed surface provided that $h$ is generic.
\endtm
We conjecture that it suffices to assume that the perturbation (which
may not be small) vanishes for $|r|\ge1$.
\pf{Proof} The reasoning in the proof of Theorem 2 leads (by (5)) to
the equation
$$
-4r^2+\e\(h_{\ph\ph}+h)|_{r=0}2\cos\ph+O(r\e)+O(\e^2)=0\tag6'
$$
of the parabolic curve. By substituting $r=\e^{1/2}\rho$ we obtain
$$
\e(-4\rho^2+2a\cos\ph+O(\e^{1/2}))=0.\tag7'
$$
The product $a\cos\ph$ (where $a=(h_{\ph\ph}+h)|_{r=0})$ has at least
eight simple roots provided that $h$ is generic (since $a$ has at
least six roots according to Sturm's theorem). The equation
$-4\rho^2+2a(\ph)\cos\ph=0$ defines at least four smooth ovals
(corresponding to the intervals where $a\cos\ph$ is positive) on the
cylinder $\{(r,\ph)\}$. The small perturbation $O(\e^{1/2})$ in
(7') does not destroy them. On $\Bbb RP^2$ we obtain at least two
parabolic ovals.\qed
\endpf
\tm{Theorem 4} Smoothing the function $w=x/(x^2+y^2)$ at the
origin followed by a generic small deformation of the resulting
surface in the projective space defines a surface with at least $4$
parabolic curves.
\endtm
\pf{Proof} Two parabolic curves appear at infinity according to
Theorem 3. We shall prove that smoothing at the origin creates
at least two another parabolic curves.
\endpf
The proof is based on the existence of a maximum point and a minimum
point of the smoothed function. Indeed, on completing the
$(x,y)$-plane by one point and applying the stereographic projection
we obtain the sphere equipped with a function that extends
$R^{-1}\cos\ph$.
This function is smooth at the point at infinity $R=\infty$ of the
sphere, and this point is noncritical for the extended function.
Generic smoothing of the singularity at the origin transforms this
extended function to a smooth function on the sphere with
nondegenerate maximum and minimum points.
In a neighborhood of the maximum point, the form $d^2F$ is negative
definite, and in a neighborhood of the minimum point, it is positive
definite. Hence, generic smoothing creates at least two parabolic
curves; one of them bounds the domain in which $d^2F$ is positive
definite and the other bounds the domain
in which $d^2F$ is negative definite.
\rk{Remark} We can readily construct an example of smoothing
creating exactly two parabolic curves near the origin.
\endrk
For smoothing of the form $F=f(R)\cos\ph$ we have
$f(R)=Rg(R^2)$. The curve of parabolic points is defined by the
equation
$$
\op{tg}^2\ph=-R^2\,\frac{f''}{f-Rf'}=3+2R^2\,\frac{g''}{g'},\quad
\text{where }\,g'=\frac{dg}{d(R^2)},
$$
as follows from formula (2). For $F=R(R^2+\e)^{-1}\cos\ph$, the
parabolic curve defined by this equation has the form of a small
lemniscate whose branches are intersecting transversally at the
singular point. A~small generic deformation merges the two hyperbolic
quadrants and generates two parabolic ovals.
\ah\bS5.\enspace Topological Properties of Gradient Mappings\endah
Consider a smooth mapping $A$ of an oriented plane onto an oriented
plane. The domain in which the Jacobian $J$ of the mapping $A$ is
positive is said to be {\it elliptic\/} and the domain where it is
negative is said to be {\it hyperbolic\/}. The elliptic and
hyperbolic domains of a generic mapping $A$ are separated by a smooth
parabolic curve $J=0$.
\ex{Example} If $A=\op{grad}F$, then the elliptic (resp., hyperbolic)
domain is the domain over which the points of the graph of $F$ are
elliptic (resp., hyperbolic).
\endex
\de{Definition} By the {\it index $i\(\ga)$} of an oriented curve $\ga$
that does not intersect the parabolic curve of the mapping $A$ we
mean the number of revolutions of the tangent vector of the curve
$A(\ga)$ (for brevity we do not indicate the mapping $A$ in this
notation).
\endde
\ex{Example} The index of the boundary of a small disk belonging to
the elliptic domain is $+1$, and that for the hyperbolic
domain is $-1$.
Let $\ga$ be a simple closed curve on the oriented plane
oriented as the boundary of the disk bounded by the curve. Suppose
that the curve does not intersect the parabolic curve.
\endex
\de{Definition} By the {\it reduced index\/} of the curve we mean the
number $\skew4\tilde i\(\ga)=i\(\ga)-\op{sign}\(J|\ga)$.
\endde
\ex{Example} The reduced index of a small simple curve is zero.
\endex
\tm{Theorem 5} The reduced index is additive. Namely, if $M$ is a
plane domain in which $J\ne0$, and if $M$ is bounded outside by a
curve $\ga_0$ and inside by curves
$\ga_j$ \rom(\<$j\ge1$\<\rom) \rom(so that $\pa
M=\ga_0-\sum\ga_j$\<\rom), then $\skew4\tilde
i\(\ga_0)=\sum\skew4\tilde i\(\ga_j)$.
\endtm
\pf{Proof} $A$ constant (translation-invariant) vector field $v$ on
the image plane can be lifted to the domain $M$ immersed in the plane
by the mapping $A$. The lift $w$ has no zeros on $M$. Denote
by $\nu_k$ and $\mu_k$ the numbers of revolutions of the tangent
vector $\dot\ga$ of the boundary and of the vector $w$ along the
component $\ga_k$ of the boundary of the domain $M$ (orienting this
component as the boundary of the disk bounded by this component).
Then $\mu_0=\sum\mu_j$ (since the field $w$ has no zeros on $M$),
while $\nu_k=1$ for all $k$. Therefore,
$$
\mu_0-\nu_0+1=\sum(\mu_j-\nu_j+1).\tag8
$$
The index of the curve $\ga$ is the number of revolutions of the
vector $A_*(\dot\ga)$ with respect to the constant field $v$. This
number differs from the number $\nu-\mu$ of the revolutions of the
vector $\dot\ga$ with respect to $w$ only by the sign $s$, namely,
$i\(\ga)=(\nu-\mu\)s$. Hence, formula (8) proves that the reduced
index $\skew4\tilde i=(\mu-\nu-1\)s$ is additive.\qed
\endpf
\de{Definition} $A$ point at which the kernel of the derivative
of the mapping $A$ is tangent to the parabolic curve is called {\it
special}.
\endde
For a generic mapping $A$, the order of tangency at a special point
is one.
\de{Definition} $A$ special point is called {\it positive}, or {\it
elliptic\/} (resp., {\it negative\/} or {\it hyperbolic\/}) if the vectors
of the kernels pointing into the elliptic domain at neighboring
parabolic points are directed from (resp., to) the parabolic point.
\endde
\ex{Example} For a gradient mapping $A=\op{grad}F$, a special point
is elliptic (resp., hyperbolic) depending on the nature of singularity of
the field obtained from the field of asymptotic directions of the
graph of $F$ by lifting this field of asymptotic direction to the
surface, in the fibered three-space of the tangent directions of the
graph, defined by the field itself (regarded as the image of a
section of the bundle of the tangent directions). This surface
(representing the field of asymptotic directions) doubly covers the
hyperbolic domain with a fold singularity of the projection over the
parabolic curve (for a generic function~$F$). The lifted direction
field turns out to be (locally) the direction field of a smooth
vector field that vanishes only over the special point. If the
special point is elliptic, then the index of this vector field at its
singular point is equal to $+1$ (i.e., this point is either a node or
a focus), and if it is hyperbolic, then it is equal to $-1$ (a~saddle
point).
Near an elliptic special point, the mapping $A$ sends the
hyperbolic half-neighborhood (in which the Jacobian is negative) to
a small domain bounded by the cusp of the caustic (as a local
diffeomorphism) and strongly stretches the elliptic
half-neighborhood. At a hyperbolic point, the hyperbolic
half-neighborhood is strongly stretched.
\endex
Consider two curves close to a parabolic curve $\ga$ in a plane,
namely, the result $\ga^+$ of a small shift of $\ga$ into the elliptic
domain and the result $\ga^-$ of a small shift of $\ga$ into the
hyperbolic domain (the three curves being oriented as the boundaries
of the corresponding disks).
\tm{Theorem 6} The difference between the indices of the curves
$\ga^+$ and $\ga^-$ is equal to the number of special points on the
parabolic curves counted according to their signs\/\rom:
$$
i\(\ga^-)-i\(\ga^+)=\si(\ga\)s(\ga),
$$
where $\si(\ga)$ is the difference between the numbers of elliptic
and hyperbolic special points, while the sign $s$ is $+1$ if the
parabolic curve is the exterior boundary of an elliptic domain\/
\rom(and $-1$ if it is the exterior boundary of a hyperbolic
domain\/\rom).
\endtm
\tm{Corollary} For the reduced indices we have $\skew4\tilde
i\(\ga^-)- \skew4\tilde i\(\ga^+)=2+\si(\ga\)s(\ga)$.
\endtm
\ex{Example} The only parabolic curve of the mapping
$A(z)={\ovz}^2+2z$ is the unit circle $|z|=1$. The three special
points are negative. In this case we have $i\(\ga^+)=1$,
$i\(\ga^-)=-2$, $\si=-3$, $s=+1$, $\skew4\tilde i\(\ga^+)=0$, and
$\skew4\tilde i\(\ga^-)=-1$.
\endex
\pf{Proof of the theorem} The curves $A(\ga^-)$ and $A(\ga^+)$ differ
only by small spirals close to the special points (that is, near the
cusps of the caustic, which is the curve of critical values of the
mapping). Each spiral contributes $\pm1$ to the index. The spiral is
placed on $\ga^+$\< if the special point is positive and on $\ga^-$
if it is negative. The contribution of the spiral to the index is equal
to $-s$. This can readily be verified by treating, for instance, the
four possible cases (and using the above description of the local
structure of the mapping in neighborhoods of elliptic and hyperbolic
special points; namely, the spiral is the image of the curve lying
in the half-neighborhood of the special point that is strongly
stretched). \qed
\endpf
Denote the boundary of a compact connected hyperbolic domain $M$ of a
gradient mapping by $\pa M=\ga_0-\sum\ga_j$ \rom(where $\ga_0$ is the
exterior boundary and the $\ga_j$ are interior boundaries, and all
curves are oriented as the boundaries of disks). Denote by $\ga^+$
the result of a small shift of the parabolic curve into the elliptic
domain.
\tm{Theorem 7} The reduced indices of the boundaries of the
hyperbolic domain of a gradient mapping that are shifted into
elliptic domains are additive, $\skew4\tilde
i\(\ga^+_0)=\sum\skew4\tilde i\(\ga^+_j)$.
\endtm
\pf{Proof} The field of asymptotic directions (i.e., zeros of the
quadratic form $d^2F$) is well defined on the double of the
hyperbolic domain. The singular points of this field are provided by
the special points of the boundary.
The Euler characteristic of the double is $2-2n$, where $n$ is the
number of interior components, $\ga_j$, of the boundary. Hence, the
sum of indices of singular points of the field on the double
is $\si(\ga_0)+\sum\si(\ga_j)=2-2n$.
However, according to the corollary to Theorem 6 we have
$\skew4\tilde i\(\ga_0^+)=\skew4\tilde i\(\ga_0^-)-2+\si(\ga_0)$
because $s(\ga_0)=-1$, and $\skew4\tilde i\(\ga^+_j)=\skew4\tilde
i\(\ga_j^-)-2-\si(\ga_j)$ because $s(\ga_j)=1$. We finally obtain
$$
\skew4\tilde i\(\ga_0^+)-\sum\skew4\tilde i\(\ga_j^+)
=\Big[\skew4\tilde i\(\ga_0^-)-\sum\skew4\tilde i\(\ga_j^-)\Big]
+\Big[2n-2+\si(\ga_0)+\sum\si(\ga_j)\Big].
$$
The first bracket vanishes by Theorem 5 and the second also vanishes,
as was proved above. \qed
\endpf
\tm{Corollary} The reduced indices of the parabolic curves of a
gradient mapping that are shifted into the elliptic domain are zero.
\endtm
\pf{Proof} By the {\it rank\/} of a parabolic curve we mean the
number of terms of the maximal (the longest) chain of parabolic curves
successively embedded in one another and starting from the given
curve (thus, the rank of a parabolic curve bounding an elliptic or
hyperbolic disk is equal to $1$). For curves of rank one bounding
hyperbolic disks, the corollary readily follows from Theorem~7
(since the interior boundary is empty).
For curves of rank one bounding elliptic disks, this follows from
Theorem 5 (because the interior boundary is empty again). Theorem 7
represents the reduced index of a parabolic curve shifted into the
elliptic domain as the sum of the reduced indices of parabolic curves
of lower rank that are shifted into the elliptic domain. This proves
the corollary by induction. \qed
\endpf
\rk{Remark} Panov and Aicardi, to whom I had communicated the above
theory, observed that for any curve $\ga$ in the elliptic domain
of a gradient mapping $A$ the numbers of revolutions of the
tangent vectors to the curves $\ga$ and $A(\ga)$ coincide. This
follows from the fact that the angle between the vectors $\dot\ga$
and $A_*\dot\ga$ never attains the values $\pm\pi/2$ (since the
scalar product $(\dot\ga,A_*\dot\ga)$ is of fixed sign in the
elliptic domain).
\endrk
\rk{Remark} Consider a gradient mapping of the plane and assume that
this mapping is hyperbolic outside a sufficiently large disk. Theorem
7 shows that no information on the intrinsic structure of elliptic
domains penetrates through the most exterior parabolic curves. The
restrictions concerning the indices and the special points provided
by the behavior of the mapping outside a large disk are related only to
the indices of the most exterior parabolic curves and to the special
points on these curves. From this point of view, the domains bounded
by these exterior curves can be regarded as elliptic domains. I do
not know whether one can really make these domains elliptic by
deforming the function $F$.
\endrk
Let us apply the above results to a generic gradient mapping of the
plane that coincides with the nongeneric mapping $A(z)=\ovz^2$ (which
has a $D_4^-$ singularity at the origin) outside a large disk.
\tm{Theorem 8} Every gradient mapping coinciding with $A$ outside a
disk centered at the origin has at least three special points on one
of its most exterior parabolic curves.
\endtm
\pf{Proof} The index of a large circle $i\(\ga_0^-)$ is $-2$. Hence,
its reduced index $\skew4\tilde i\(\ga_0^-)$ is $-1$. By Theorem~5,
it is equal to the sum of the reduced indices of hyperbolic shifts of
the exterior parabolic curves $\ga_j$, and hence $\sum\skew4\tilde
i\(\ga_j^-)=-1$. According to the corollary of Theorem 6, we have
$\skew4\tilde i\(\ga_j^-)=\skew4\tilde i\(\ga_j^+) +2+\si(\ga_j)$
(\<$s(\ga_j)=1$ because this parabolic curve is the exterior boundary
of an elliptic domain). By the corollary of Theorem 7 we have
$\skew4\tilde i\(\ga_j^+)=0$. Thus, $\sum(2+\si(\ga_j))=-1$. Hence,
at least one of the summands is negative. For the sum of the signs of
the special points on the exterior parabolic curve that corresponds
to this summand we have the inequality $\si(\ga_j)\le-3$. Therefore,
there are at least three special points on this parabolic curve.\qed
\endpf
\rk{Remark} Moreover, we have proved that there is an exterior
parabolic curve on which the number of hyperbolic special points
exceeds that of elliptic ones at least by three. This bound is
sharp: for the gradient deformation $A(z)=\ovz^2+\e z$ (provided by
the versal deformation of the $D_4^-$\< singularity), there are
exactly three special points (exactly three cusps on the caustic).
\endrk
One may conjecture (in accordance with the general principle of
topological minimality for algebraic objects) that similar
inequalities hold for the other boundary conditions as well, namely,
the set of singularities into which a degenerate gradient Lagrangian
singularity is decomposed under a small generic gradient deformation
that makes this singularity generic, is possibly topologically
necessary (cannot be reduced by a different choice of a gradient
deformation).
For instance, in the two-dimensional case, consider the number of
cusps of the caustic of a gradient mapping obtained by a continuation
of a potential function (defining a degenerate singularity at the
origin) from a small annulus centered at the origin to the interior
disk. One may conjecture that the minimal value of this number of
cusps is attained under a generic small deformation of the potential
function, and that the other continuations, however sophisticated,
provide a larger number of cusps.
\ex{Example} The degenerate singularity of the gradient mapping
$z\mapsto\ovz^m$ has a standard small gradient deformation
$z\mapsto\ovz^m+\e z$ with a single parabolic curve carrying $m+1$
(hyperbolic) special points. {\it There exists no deformation\/
\rom(vanishing outside a disk centered at the origin\/\rom) whose
caustic has fewer cusps}.
\endex
\pf{Proof} As in Theorem 8 we see that $i(\ga_0)=-m$, $\skew4\tilde
i\(\ga^-_0)=1-m$, $\sum\skew4\tilde i\(\ga_j^-)=1-m$, and
$\skew4\tilde i\(\ga_j^-)=\skew4\tilde i\(\ga_j^+)+2+\si(\ga_j)$. If
$-\si(\ga_j)<m+1$ for every exterior boundary, then $\sum\skew4\tilde
i\(\ga_j^-) =\sum(2+\si(\ga_j))>\sum(2-m-1)\ge1-m$, which contradicts
the boundary condition $\sum\skew4\tilde i(\ga_j^-)=1-m$. Thus, there
exists an exterior parabolic curve for which $\si(\ga_j)\le-1-m$.\qed
\endpf
Let us apply the above results to the study of the compactly
supported deformations of a surface that is hyperbolic at infinity
(over the M\"obius band complementary to the disk on which the
deformations are concentrated) and is transversal to the vertical
direction.
An example of such a surface at infinity is defined by the function
$$
H(r,\ph)=\cos3\ph+r\sin2\ph-cr^2\cos\ph,\qquad0<c<1/4,\tag9
$$
in the polar coordinates of \S2.
We can readily verify the following properties of the field of
asymptotic directions of a surface hyperbolic at infinity (we project
the field to the M\"obius band along the vertical direction and
consider it as a field defined on the M\"obius band).
1. {\it The two fields of asymptotic directions on the M\"obius band
are distinguishable\/}: one of them is orientable while the other is
not.
2. {\it The number of the tangency points of asymptotic direction on
the M\"obius band with the central line at infinity is at least
three}, where the number of the points of tangency with the
orientable field is odd and with the nonorientable is even.
This follows from the M\"obius theorem, since the tangency points are
exactly the projections of the inflection points of the line of
intersection of the surface with the vertical projective plane
containing the line at infinity.
There are three such inflection points provided that the plane
intersects the surface transversally along a smooth noncontractible
curve (the perturbation can be so large that the perturbed surface
can be tangent to some vertical lines).
3. Consider the torical surface fibered into circles over the double
covering of the line at infinity, where the fibers consist of the
unit vectors tangent to the cylinder that doubly covers, of the
M\"obius band at the central circle points (covering the points of
the line at infinity). This bundle is induced by the restriction of
the unit tangent vector bundle of the M\"obius band by the mapping of
the circle onto the line at infinity covering it twice.
Each of the two fields of asymptotic directions along the line at
infinity can be lifted to a section of the above bundle. {\it The
intersection indices of the resulting curves on the torus with the
section obtained by the lifting of the direction of the line at
infinity are zero}.
4. To describe the rotation type of both asymptotic directions at
infinity, it is convenient to use cyclic words of length
$2\(2k+1)$ (\<$2k+1$ being the total number of tangencies of both
fields with the direction of the line at infinity). The four symbols
of the alphabet describing the rotations are
$\{\cup,\cap,\dup,\ddn\}$.
Each symbol describes the rotation of a frame of asymptotic
directions between two consecutive points of tangency of the
asymptotic field with the central circle (on the cylinder, which
doubly covers the M\"obius band). The symbol ``$\dup$'' (``$\ddn$'')
stands for a positive (negative) rotation of the frame (the
orientation of the cylinder being transported from the affine part of
the projective plane). The grammar rules reflect the continuity of
the rotation: ``$\cup$'' can be followed only by ``$\cap$'' and
``$\dup$'', ``$\cap$'' only by ``$\cup$'' and ``$\ddn$'', ``$\dup$''
only by ``$\cap$'' and ``$\dup$'', ``$\ddn$''only by ``$\cup$'' and
``$\ddn$''. The cyclic words are supposed to be odd (the opposite
symbols being opposite), and the number of the ``$\cup$''s (and hence
of the ``$\cap$''s) should be odd.
Since our words are cyclic, the words obtained from one another by
the involution replacing each symbol by the opposite symbol can be
identified. The symmetry allows us to represent our cyclical words by
their halves, which I shall call {\it snakes}. Each snake is a word
of $2k+1$ symbols, the total number of the ``$\cup$''s and
``$\cap$''s in a snake must be odd, and the snake produces a cyclic
word of length $2\(2k+1)$ if we follow it by a snake of opposite
symbols. Opposite snakes represent the same cyclic word.
A snake is said to be {\it admissible\/} if the corresponding cyclic
word satisfies the above rules, and we do not distinguish the snakes
representing the same cyclic word.
For instance, there are only two admissible snakes of length $3$,
namely, $\cap\cup\cap$ and $\cap\ddn\ddn$. The numbers of admissible
snakes of lengths $1,3,5,7,9,11,\dots$ form a strange sequence
$1,2,4,10,30,94,\dots$. These numbers count the cyclically ordered
partitions of an odd number $n$ into an odd number of summands.
\rk{Remark} The numbers of unordered partitions of an odd number
$n=2k+1$ into an odd number of summands form a sequence
$1,2,4,8,16,29,\dots$ (\<$k=0,1,2,3,4,5,\dots$), which is even more
strange. On classifying the partitions by the number of the summands,
we obtain a triangle for which I could not find a general rule:
$$
\gather
1\\
\vspace{-3pt}1\quad1\\
\vspace{-3pt}1\quad2\quad1\\
\vspace{-3pt}1\quad4\quad2\quad1\\
\vspace{-3pt}1\quad7\quad5\quad2\quad1\\
\vspace{-3pt}
1\quad\kern-2.5pt10\kern-2.5pt\quad\kern-2.5pt10\kern-2.5pt\quad5\quad2
\quad1
\endgather
$$
\endrk
It seems that the admissible snakes are realizable by the fields of
asymptotic directions of some surfaces that are hyperbolic at
infinity, but I have not proved this. I do not know how a snake at
infinity restricts the field of asymptotic directions at the
parabolic curves over an affine disk for a surface which is
hyperbolic at infinity. I think that computer experiments can be
useful to obtain better understanding of this problem.
Passing from the M\"obius band (forming a neighborhood of the line at
infinity) to the complementary affine disk, we obtain the following
boundary conditions for the field of asymptotic directions
(\<$d^2F=0$) of the corresponding gradient mapping
(\<$A=\op{grad}F$).
A) {\it The number of revolutions, of any of the two asymptotic
directions, performed while the observation point makes one
revolution along the boundary of a large disk on the affine plane is}
$+1$ (provided that the surface is hyperbolic at infinity). In
particular, {\it both fields of asymptotic directions are orientable
over this large circle.}
B) {\it The number $i\(\ga_0^-)$ of revolutions of the tangent vector
to the image $A\ga_0^-$ of a large circle, under the gradient
mapping defined by a surface hyperbolic at infinity, is $+1$}.
C) {\it The number of revolutions of the acceleration vector
$d^2A\ga_0^-(\ph)/(d\ph)^2$ performed while the observation point
makes one revolution along the boundary of a large disk is at least
four.}
For our special example of the surface (9), this number of
revolutions is four. In this example, the curve $A\ga_0^-$ described
by the image of the point moving along the circle $R=\op{const}$
bounding a large disk is a small deformation of a hypocycloid with
three cusps passed twice in the negative direction. The deformation
replaces three consecutive cusps by small spirals and smoothes the
next three cusps. The tangent vector to the deformed curve makes
$-2+3=+1$ revolutions as the argument of the point running along the
large circle changes from $0$ to $2\pi$.
For cubic surfaces, the number of parabolic curves is equal to four
and the number of special points is equal to six, and all these
points are negative. This agrees with our above formulas,
$$
\skew4\tilde i\(\ga_0^-)=\sum\skew4\tilde i\(\ga_j^-)
=\sum(2+\skew4\tilde i\(\ga_j^+)+\si(\ga_j))=8-6=2=i\(\ga_0^-)+1.
$$
\rk{Remark} It seems that a large positive value of the index
$i\(\ga)$ (where $\ga$ is a curve in the hyperbolic domain bounding a
disk containing the parabolic curves) implies the existence of many
exterior parabolic curves of a given gradient mapping. This would
follow if the Panov conjecture were true; this conjecture claims
that, for a parabolic curve bounding a disk whose exterior points
(close to the parabolic curve) are hyperbolic, the index $i\(\ga^-)$
of a small shift of the parabolic curve into the hyperbolic domain
that is exterior for this curve is at most one. Panov stated this
conjecture only for void parabolic curves bounding elliptic
disks, but I know no counterexample even to the more general
conjecture formulated above. This conjecture seems to be related to
the following circumstances:
1) the index of a singular point of a plane gradient field is at most
one;
2) the relation between the function and the direction of its
gradient is invariant under the conformal diffeomorphisms;
3) every disk is conformally equivalent to the standard one;
4) all special points on a convex parabolic curve bounding an
elliptic domain are negative.
Of course, these arguments do not prove
the conjecture, but the attempt to prove it by contracting the
parabolic curve to a singular point of the surface seems to be
natural.
\endrk
\Refs
\item{1.} B. Segre, The Nonsingular Cubic Surfaces, Clarendon Press,
Oxford, 1942.
\item{2.} A. Hurwitz, ``\"Uber die Fourierschen Konstanten
integrierbarer Funktionen,'' Math. Ann., {\bf57}, 425--446 (1903).
\item{3.} V. I. Arnold, ``Topological problems of the theory of wave
propagation,'' Usp. Mat. Nauk, {\bf51}, No.~1, 3--50 (1996).
\endRefs
\transl{V. I. Arnold}
\end
