There are several algorithms (ShapeDNA , Heat Kernel Signature , Wave Kernel Signature , etc.) based on eigenvalues and associated eigenfunctions of the Laplace-Beltrami operator
to construct spectral shape descriptors that depend on mesh surface.
Among the topics are escape from large holes in Anosov systems, isolated elliptic fixed points for smooth Hamiltonians, thermodynamic formalism for some systems with countable Markov structures, spectral boundary value problems for the Laplace-Beltrami operator
: moduli of continuity in eigenvalues under domain deformation, and measure-theory properties of center foliations.
Let [H.sup.d] be the d-dimensional hyperbolic space and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the Brownian motion on [H.sup.d] generated by the half of the Laplace-Beltrami operator
. For a fixed point o [member of] [H.sup.d], define P = [P.sub.o] and [R.sub.t] = d (o, [X.sub.t]), where d is the distance function of [H.sup.d].
According to [23, 29], the discretization of the Laplace-Beltrami operator