This paper is an exposition of some results concerning two distinct mean curvature equations which have been obtained recently by the author and others. We also propose some unsolved problems and new questions arising from ours developments.

A hyperbolic flow by mean curvature equation, l t #cv"i, for the evolution of interfaces is studied. Here v, i and l t are the normal velocity, curvature and normal acceleration of the interface. A crystalline algorithm is developed for the motion of closed convex polygonal curves; such curves may exhibit damped oscillations and their shape appears to rotate during the evolutionary process. The...

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss a...

1. Let M and M' denote complete riemannian manifolds of dimension n and m respectively, and suppose that M is compact and oriented. For simplicity we assume that both manifolds and their metrics are smooth (i.e. of class C). In terms of local co-ordinates (x, x, ...,x") on M and local co-ordinates (y,y, •••,/") on M', the riemannian metrics are written ds = gtJ dx l dx, ds' = g'aP dy* dy * wher...