Hermann Karcher

The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions

We prove existence of Schoens and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant mean curvature surfaces by solving analogous Plateau problems in S3. The required contours in S3 are obtained by working with the great circle orbits of Hopf S1-actions in the same way as with families of parallel lines in ℝ3. Annular Plateau problems give new embedded minimal surfaces in S3. For many of the minimal surfaces in ℝ3 global Weierstraß representations are derived.

Embedded minimal surfaces derived from Scherk's Examples.

SummaryIn this article we construct embedded minimal surfaces which are, at least heuristically, derived from Scherks first and second surface. Our examples are either parametrized by punctured spheres and then have one translational period or one screw motion period; or they are parametrized by rectangular tori and then have one or two translational periods. The helicoidal examples contain nonisometric ∈-deformations in the sense of Rosenberg [R].

Geometrische Methoden zur Gewinnung von A-Priori-Schranken für harmonische Abbildungen

In this paper, we prove a-priori estimates for harmonic mappings between Riemannian manifolds which solve a Dirichlet problem. These estimates employ geometrical methods and depend only on geometric quantities, namely curvature bounds, injectivity radii, and dimensions. An essential tool is the introduction of almost linear functions on Riemannian manifolds. Furthermore, we show the existence of almost linear and harmonic coordinates on fixed (curvature controlled) balls. These coordinates possess better regularity properties than Riemannian normal coordinates.

Construction of Triply Periodic Minimal Surfaces

We discuss triply periodic minimal surfaces from a mathematical point of view, giving concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directions for further generalizations.

Adding handles to the helicoid

There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one

Anwendungen der Alexandrowschen Winkelvergleichssätze

A generalized version of Alexandrows angle comparison theorems is stated (1) and the following applications are given: A new proof of Klingenbergs estimate of the cut locus distance and related equality discussions (2), an existence proof for geodesic loops and for one closed geodesic (3), a new proof for the convexity of metric balls (4), a lemma concerning approximation of convex curves by polygons (5), lower and (known) upper bounds for the length of convex curves in terms of their geodesic curvature and the Gaussian curvature (6) and another comparison theorem for geodesic triangels (7).

A Family of Singly Periodic Minimal Surfaces Invariant under a Screw Motion

We construct explicitly, using the generalized Weierstrass representation, a complete embedded minimal surface M k ,θ invariant under a rotation of order k + 1 and a screw motion of angle 2θ about the same axis, where k > 0 is any integer and ois any angle with |θ| < π/(k + 1). The existence of such surfaceswas proved in [Callahan et al. 1990), but no practical procedure for constructing them was given there. We also show that the sameproblem for θ = ±π/(k+1) does not have a solution enjoying reflective symmetry; the question of the existence of a solution without such symmetry is left open.

On Shikata's distance between differentiable structures

Shikata proved: there is a number ε(n) with the following property: If two compact homeomorphic n-dimensional manifolds have a distance less than ε (n), then they are diffeomorphic. We improve the known lower bound ∼(n!)−n for ε(n) to ∼1/3n−2.