  Modeling of a wide class of physical phenomena, such as crystal growth and
flame propagation, leads to tracking fronts moving with curvature-dependent
speed. When the speed is the curvature this leads to one of the classical
degenerate nonlinear second order differential equations on Euclidean space.
One naturally wonders "what is the regularity of solutions?" A priori solutions
are only defined in a weak sense, but it turns out that they are always twice
differentiable classical solutions. This result is optimal; their second
derivative is continuous only in very rigid situations that have a simple
geometric interpretation. The proof weaves together analysis and geometry.
Without deeply understanding the underlying geometry, it is impossible to prove
fine analytical properties.
