Regular homotopy and total curvature

Abstract

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8\pi and we establish a non-injectivity result for local minima.Comment: This is the version published by Algebraic & Geometric Topology on 23 March 2006. arXiv admin note: this version concatenates two articles published in Algebraic & Geometric Topolog