Conservation Laws and Formation of Singularities in Relativistic Theories of Extended Objects

The dynamics of an M-dimensional extended object whose M+1 dimensional world volume in M+2 dimensional space-time has vanishing mean curvature is formulated in term of geometrical variables (the first and second fundamental form of the time-dependent surface $\sum_M$), and simple relations involving the rate of change of the total area of $\sum_M$, the enclosed volume as well as the spatial mean -- and intrinsic scalar curvature, integrated over $\sum_M$, are derived. It is shown that the non-linear equations of motion for $\sum_M(t)$ can be viewed as consistency conditions of an associated linear system that gives rise to the existence of non-local conserved quantities (involving the Christoffel-symbols of the flat M+1 dimensional euclidean submanifold swept out in ${\Bbb R}^{M+1}$). For M=1 one can show that all motions are necessarily singular (the curvature of a closed string in the plane can not be everywhere regular at all times) and for M=2, an explicit solution in terms of elliptic functions is exhibited, which is neither rotationally nor axially symmetric. As a by-product, 3-fold-periodic spacelike maximal hypersurfaces in ${\Bbb R}^{1,3}$ are found.