Remarks on the Parabolic Curves on Surfaces
and on the Many-Dimensional M\"obius--Sturm Theory

V. I. Arnold

August 26, 1997

tAccording 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 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 for
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) is four.  This leads to the Aicardi conjecture:
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 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).

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.  Possibly the most natural approach is to consider the polyhedral
deformations constructed from the triangulations of the perturbed
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
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 $F_{xx}F_{yy}=F^2_{xy}$.