Surface triangulations with isometric boundary

AbstractLet T be a triangulation of a bordered compact surface, and let C be a boundary component of T. Consider the metric on V(T) as determined by the 1-skeleton of T. T is isometric with respect to C if for any two vertices of C their distance in T is equal to the distance on C. Let n be the number of vertices on C, and assume that the number of vertices on all other boundary components of T is o(n). If T is an isometric triangulation of the disk with holes then |V(T)|=Ω(n2). T is irreducible if the contraction of any interior edge results in a nonisometric triangulation or changes the homeomorphism type of the surface. It is shown that the number of combinatorially distinct irreducible isometric triangulations of a fixed surface with n vertices on the boundary is finite for each n