We consider maintaining the contour tree T of a piecewise-linear
triangulation M that is the graph of a time varying height function
h:R2→R. We carefully describe the
combinatorial change in T that happen as h varies over time and
how these changes relate to topological changes in M. We present a
kinetic data structure that maintains the contour tree of h over time. Our
data structure maintains certificates that fail only when h(v)=h(u) for two
adjacent vertices v and u in M, or when two saddle vertices lie
on the same contour of M. A certificate failure is handled in
O(log(n)) time. We also show how our data structure can be extended to
handle a set of general update operations on M and how it can be
applied to maintain topological persistence pairs of time varying functions