Tight immersions of surfaces in \(E^ 3\) (Q1199238)

This paper solves a long-standing problem posed by \textit{N. H. Kuiper} in Comment. Math. Helv. 35, 85-92 (1961; Zbl 0243.53043): Does there exist a tight immersion of the projective plane with one handle into \(E^ 3\)? In other words: Does there exist an immersion of this surface with total absolute curvature \(\int| K|=2\pi\cdot 5\)? The main result (Théorème A) says that there is no smooth immersion of this kind. Besides standard techniques like critical point theory, top-cycles, singularities of excellent (2-generic) mappings, the proof involves a new idea developed by the author. This concerns the global structure of the profile curves associated with linear projections of the surface to the plane, namely the part of the surface which is folded by such a projection. Assuming the existence of a tight surface with \(\chi=-1\), a careful combination of these methods leads to a contradiction.