Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature
For researchers in geometric modeling and numerical analysis, this provides a general convergence theory for nonlinear subdivision on curved spaces, significantly extending prior work.
This paper proves that subdivision schemes on Riemannian manifolds with nonpositive curvature converge if and only if their linear counterparts converge, removing previous sign restrictions on the mask and applying to all input data. The result establishes almost equivalence between convergence of linear and nonlinear manifold schemes.
This paper studies well-defindness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian center of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that convergence is implied by contractivity of a derived scheme, resp. iterated derived scheme. In this way we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.