Abstract

We consider the following question: among all curves lying on a given Riemannian manifold with prescribed length and boundary data, which ones minimise the $L^\infty$ norm of the curvature? This extends a paper of Moser considering the same question in Euclidean space. Using the method of $L^p$ approximation we show that minimisers of our problem and also a wider class of “pseudominimiser” curves must satisfy an ODE system obtained as the limit as $p \rightarrow \infty$ of the $L^p$ Euler-Lagrange equations. This system gives us some geometric information about our (pseudo)minimisers. In particular we find that their curvature takes on at most two values: a positive constant $K$, and possibly zero in some places.

This talk is based on joint work with Roger Moser which can be found at https://arxiv.org/abs/2202.07407.