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...

We prove the existence of (branched) conformal immersions F : S → R with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S, H and partial results for surfaces of higher genus.

Let M be a closed hyperbolic three manifold. We construct closed surfaces that map by immersions into M so that for each, one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.

1 Geometry of Manifolds 3 1.1 Definition of smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Tangent space to a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Smooth functions on manifolds, partitions of unity. . . . . . . . . . . . . . . 22 1.5 Immers...

Using generating functions, we derive a multiple point formula for every generic immersion f : M # N between even dimensional oriented manifolds. This produces explicit formulas for the signature and Pontrjagin numbers of the multiple point manifolds. The formula takes a particular simple form in many special cases, e.g. when f is nullhomotopic we recover Szűcs’s formula in [3]. It also include...

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...