Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

TL;DR

This paper investigates the convergence and well-definedness of subdivision schemes on Riemannian manifolds with nonpositive curvature, extending previous work by removing sign restrictions and analyzing the continuity of limit curves.

Contribution

It introduces a general framework for convergence of nonlinear subdivision schemes on manifolds, linking linear scheme convergence to manifold scheme convergence without sign restrictions.

Findings

Convergence is guaranteed if a derived scheme is contractive.

The convergence of linear schemes is nearly equivalent to that of their nonlinear counterparts.

The paper establishes conditions for H"older continuity of limit curves.

Abstract

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 H\"older 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.