A note on the regularity and the existence of Riemannian splines

TL;DR

This paper provides a detailed proof of the regularity of Riemannian splines' critical points and establishes the existence of minimizers for spline energy functionals with multiple interpolation points.

Contribution

It offers the first comprehensive proof of regularity for higher-order Riemannian splines and demonstrates existence of minimizers with multiple interpolation points.

Findings

Proof of regularity for critical points of spline energy on Riemannian manifolds

Existence of minimizers with multiple interpolation points

Generalization of DuBois-Reymond Lemma for this context

Abstract

In this paper, we present a comprehensive proof concerning the regularity of critical points for the spline energy functional on Riemannian manifolds, even for the general higher-order case. Although this result is widely acknowledged in the literature, a detailed proof was previously absent. Our proof relies on a generalization of the DuBois-Reymond Lemma. Furthermore, we establish the existence of minimizers for the spline energy functional in cases where multiple interpolation points are prescribed alongside just one velocity.