The asymptotic blow-up of a surface in Euclidean 3-space (Q1179139)

The asymptotic directions of a smoothly immersed surface in \(\mathbb{R}^ 3\) define a subset in the Grassmann bundle of unoriented 1-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. This can be used to show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface. Furthermore, an existence theorem for surfaces which satisfy regularity conditions and a symplectic Monge-Ampère PDE is derived.