Stable planes

Endomorphisms of Stable Planes

Endomorphisms of stable planes are introduced, and it is shown that these are injective, locally constant or collapsed. Examples are studied, and it is shown that there are stable planes admitting \substantially more" endomorphisms than automorphisms. It seems to be a rather popular feeling that geometry is the domain of groups and symmetry, whereas asymmetry and semigroups have been banished t...

Solvable Groups of Automorphisms of Stable Planes

An interesting problem in the foundations of geometry is the following question: What is the impact of the interplay of topological assumptions and homogeneity? One possibility to make this (rather philosophical) question treatable for a mathematician is the classiication project for stable planes. We shall brieey outline the necessary deenitions and basic (though occasionally deep) results. In...

Approximating Stable Sets Using the #-fun tion and Cutting Planes

Stable Complete Minimal Surfaces in R Are Planes

A proof of the statement in the title is given.

Stable Complete Embedded Minimal Surfaces in H1 with Empty Characteristic Locus Are Vertical Planes

In the recent paper [12] we have proved that the only stable C minimal surfaces in the first Heisenberg group Hn which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in [12] to C complete embedded minimal surfaces in H with empty cha...