Regular Homotopy and Total Curvature

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 also consider the total curvature functional on the space of 2-sphere immersions into 3-space in a similar spirit. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8π and we establish a non-injectivity result for local minima. Recently, A. Borbély [3] showed that any sphere eversion has an instant when the total curvature is at least 8π and thus the infimum equals 8π.