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\input amstex
\documentstyle{amsppt}
\input epsf

\def\R{\Bbb R}
\def\RP{\Bbb{RP}}
\def\C{\Bbb C}
\def\eps{\varepsilon}
\def\Vol{\operatorname{vol}}
\def\volume{\operatorname{volume}}

\rightheadtext{ Sharpness of Risler's bound for the total curvarure }

\topmatter
\title     Sharpness of Risler's upper bound for the total
            curvature of an affine algebraic hypersurface
\endtitle
\author     S.Yu.~Orevkov
\endauthor
\address    Steklov Math. Inst., Moscow, Russia
\endaddress
\address    Lab. E. Picard, Univ, Paul Sabatier, Toulouse, France
\endaddress
\endtopmatter



\document

Let $A$ be a real algebraic hypersurface in $\R^{n+1}$ of degree $d$.
For a point $p\in A$, we denote the curvature of $A$ at $p$ 
(i.e., the jacobian of the Gauss mapping $\gamma:A\to S^{n}$) by 
$k(p)$.
Similarly, for a point $p$ in the
complexification $\C A$ of $A$, we denote the Gaussian curvature of $\C A$ at $p$ by $K(p)$.
Using results of Teissier and Langevin, Risler [1] proved that
$$
 {1\over\sigma_{n}}\int_{A} |k| \le 
 {1\over\sigma_{2n}}\int_{\C A} |K| = {d(d-1)^n\over 2}
       \qquad\text{("$<$" instead of "$\le$" for $d\ge 3$)}
                  \eqno(1)
$$
where $\sigma_n$ is the $n$-volume of the unit $n$-sphere.
Using Harnack's construction he proved also
that this bound is sharp for $n=1$, i.e., for any $d$ and for any $\eps>0$
there exists a real algebraic curve $A$ in $\R^2$ such that 
$\int_{A}|k|>\pi d(d-1)-\eps$. The purpose of this note is to show
the evidence of the fact that Risler's bound (1) is sharp for any $n$.
Namely,

\proclaim{ Proposition }
For any positive integers $n$, $d$ and for any $\eps>0$ there exists
a real algebraic hypersurface $A$ in $\R^{n+1}$ such that 
$ \int_{A} |k| > {1\over 2}d(d-1)^n\sigma_n - \eps$.
\endproclaim

\demo{ Proof } Let $f$ be a polynomial in one variable of degree $d$
which has $d$ distinct real roots $a_1,\dots,a_d$ in the segment $[-1,1]$.
For $c\in\R$, let us set 
$$
 F_c(x_0,\dots,x_n) = f(x_0) + c\cdot\big(f(x_1) +\dots+ f(x_n)\big),
$$
and let $A_c$ be the hypersurface in $\R^{n+1}$ defined by the equation $F_c=0$.
\medskip

{\bf Step 1.} {\it There exists $\delta>0$ such that $0<|c|<\delta$ implies
that the restriction of $x_0$-coordinate to the hypesurface $A_c$ has
$d(d-1)^n$ nondegenerate critical points.}
\smallskip

Indeed, critical points of $x_0$ on $A_c$ are solutions
of the simultaneous equations
$$
    {\partial F_c\over\partial x_1}=\dots
    ={\partial F_c\over\partial x_n}=F_c=0                    \eqno(2)
$$
By Rolle's theorem, $f'$ has $d-1$ real roots $b_1,\dots,b_{n-1}$.
Thus, if $c\ne 0$ then
the solution of the simultaneous equations $\partial F_c/\partial x_j=0$,
$j=1,\dots,n$, is the union of $(d-1)^n$ lines $x_j=b_{i_j}$, $1\le i_j\le n$.
The $d$ hyperlanes $x_0=a_i$, $i=1,\dots,d$ are transverse to these lines,
so if we include the equation $F_0=0$, we obtain $d(d-1)^n$ solutions.
It is clear that for $|c|\ll 1$, the hypersurface $F_c=0$ is $C^1$-close
to $F_0=0$ in the cube $[-1,1]^{n+1}$, so the perturbed system of equations
has the same number of solutions. Moreover, these solutions are nondegenerate,
hence they yield nondegenerate critical points.

\medskip
{\bf Step 2.} {\it Let us fix $c$, $0<|c|<\delta$, and let us set 
$A_{c,h}=\{F_{c,h}=0\}$ where
$$
     F_{c,h}(x_0,\dots,x_n) = F_c(hx_0,\, x_1,\dots,x_n).
$$
Then one has $\;\lim_{h\to 0}\int_{A_{c,h}}|k| \ge {1\over2}d(d-1)^n\sigma_n$.}
\smallskip

Indeed, it is clear that $A_{c,h} = s_h(A_c)$ where $s_h:\R^{n+1}\to\R^{n+1}$ is the
stratching $(x_0,\dots,x_n)\mapsto(x_0/h,\,x_1,\dots,x_n)$.
Let $p_1,\dots,p_N\in A_c$ (where $N=d(d-1)^n$) 
be the critical points of $x_0|_{A_c}$.
Let us choose disjoint neighbourhoods $U_1,\dots,U_N\subset A_c$ of the points
$p_1,\dots,p_N$. It is enough to prove that 
$\;\lim_{h\to 0}\int_{s_h(U_\nu)}|k| \ge \sigma_n/2$ for any $\nu$.
By definition, $k$ is the jacobian of the Gauss map
$\gamma_h: A_{c,h}\to\RP^n$ with respect to the metric on $\RP^n$
induced by the standard projection of the unit sphere $S^n$ onto $\RP^n$. 
Hence, $\int_{s_h(U_\nu)}|k| \ge \Vol_{\RP^n} \gamma_h(s_h(U_\nu))$.
Let us consider the affine chart $V_0$ on $\RP^n$ corresponding to the 
coordinates $X_1,\dots,X_n$ where $X_i=x_i/x_0$. The fact that $p_\nu$ is a
nondegenerate critical point means that $\gamma_1(U_\nu)$ contains an open
ball $B\subset V_0$ centered at the origine.
It remains to note that $\gamma_h(s_h(U_\nu)) = S_h(\gamma_1(U_\nu))$ where
$S_h:V_0\to V_0$ is the homothety $(X_1,\dots,X_n)\mapsto(X_1/h,\dots,X_n/h)$,
hence
$$
 \Vol_{\RP^n} \gamma_h(s_h( U_\nu )) \ge \Vol_{\RP^n} S_h(B)
 \underset{h\to0}\to\longrightarrow \volume(\RP^n) = \sigma_n/2. \qed
$$
\enddemo

\medskip

\remark{ Remark } In fact, Risler proved in [1] more than the sharpness of (1) for $n=1$.
He proved that (1) for $n=1$ remains sharp if one considers only {\it maximal curves}, 
i.e. curves which have the maximal possible number $(d-1)(d-2)/2+d$ of connected 
components (recall that the curves are affine). 
Our construction applied for $n=1$ provides curves
which are far from being maximal.
\endremark


\Refs
\ref\no 1\by J.-J.~Risler 
\paper On the curvature of the real Milnor fiber
\jour Bull. London Math. Soc. \vol 35 \yr 2003 \pages 445-454
\endref

\endRefs
\enddocument
