Sharpness of Risler's upper bound for the total curvature of an affine algebraic hypersurface

by  S.Yu. Orevkov


We give a short proof of the sharpness of the following inequality proved in [ J.-J.Risler. On the curvature of the real Milnor fiber, Bull. London Math. Soc., 35(2003), 445-454]: $\int_{RA}|k| < \int_{CA}|K|$ where  RA  is an affine real hypersurface in $R^n$;  CA  its complexification,  k  and  K  are curvatures of  RA  and  CA  properly rescaled.