Ein verschärfter und verallgemeinerter Satz von Alexandrov

In this paper we prove the following theorem: Suppose that n≥3 and 1≤jn $$(\forall a,b) d(a,b) : = \sum\limits_{\nu = 1}^j { (a_\nu - b_\nu )^2 - \sum\limits_{\nu = j + 1}^n { (a_\nu - b_\nu )^2 .} }$$ If a function f:?ⁿ→?ⁿ satisfies the condition: (*) $$(\forall x,y \in \mathbb{R}^n ) d(f(x),f(y)) = 0 \Leftrightarrow d(x,y) = 0,$$ then f is affine. Moreover, f preserves distances up to a constant factor C≠0, i.e. d(f(x),f(y))=C·d(x,y) for every x,y. In contrast to Alexandrov's result [1] we do not assume that f is bijective, and we also do not assume that j=n?1. A very important part of our proof will be the discussion of a functional equation.     

Plateau's problem for surfaces of prescribed mean curvature in given regions

Plateau's problem for surfaces of prescribed mean curvature will be considered in regions which are not necessarily H-convex.     

Chernklassen semi-stabiler Rang-3-vektorbündel auf kubischen Dreimannigfaltigkeiten

It is shown that the third Chern-number of a semistable Rk-3-vector bundle on a smooth hypersurface of degree 3 in ₄ can be bounded by the first and the second Chern-number of the bundle.