Matthias Weber

Teichmuller theory and handle addition for minimal surfaces

We develop Teichmuller theoretical methods to construct new minimal surfaces in

An embedded genus-one helicoid

\BE^3

Period quotient maps of meromorphic 1-forms and minimal surfaces on Tori

by adding handles and planar ends to existing minimal surfaces in

The construction of doubly periodic minimal surfaces via balance equations

\BE^3

On properly embedded minimal surfaces with three ends

. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gau\ss map for their genus; the (Weierstrass data) period problem for these surfaces is of arbitrary dimension. In particular, we exhibit a two-parameter family of complete minimal surfaces in the Euclidean three-space

A Teichmüller theoretical construction of¶high genus singly periodic minimal surfaces invariant¶under a translation

\BE^3

Complete Embedded Harmonic Surfaces in R3

which generalize the breakthrough minimal surface of C. Costa; these new surfaces are embedded (at least) outside a compact set, and are indexed (roughly) by the number of ends they have and their genus. They have at most eight self-symmetries despite being of arbitrarily large genus, and are interesting for a number of reasons. Moreover, our methods also extend to prove that some natural candidate classes of surfaces cannot be realized as minimal surfaces in

Construction of harmonic surfaces with prescribed geometry

\BE^3

Handle addition for doubly-periodic Scherk surfaces

. As a result of both aspects of this work, we obtain a classification of a family of surfaces as either realizable or unrealizable as minimal surfaces.

About the cover: Early images of minimal surfaces

There exists a properly embedded minimal surface of genus one with a single end asymptotic to the end of the helicoid. This genus-one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces, also asymptotic to the helicoid, that have genus equal to one in the quotient.