Minimising Maximum Curvature

: (Alternative Format Thesis)

Abstract

In this thesis we study two geometric variational problems in L∞. Firstly, we look at minimising the L∞ norm of the geodesic curvature (crudely speaking, minimising the maximum curvature) of curves with prescribed boundary data and length on a Riemannian manifold, generalising a previous result of Moser [63] which applies to curves in Euclidean space. Secondly, we look at the novel problem of minimising the weighted L∞ norm of the mean curvature of topological spheres immersed in R3.

Utilising the method of Lp approximation, we construct our minimisers as the sequential limit of a sequence of Lp minimisers as the parameter p tends to infinity. At the same time, we study the Euler-Lagrange equations for the Lp minimisers, showing that the equations converge as p → ∞ to a limiting system of differential equations satisfied by a vector field or function, in the case of curves and surfaces respectively, defined on the L∞ minimiser.

We show by including a penalisation term in our analysis that all L∞ minimisers, as well as elements of a larger set analogous in a sense to critical points, known as “pseudominimisers,” must satisfy such a limiting system of equations. For both problems, we analyse the system and find a connection between its solution and the curvature of the curves/surfaces: the curvature is zero on the zero set of the system’s solution, and the (weighted) curvature has constant magnitude away from the zero set.

We conclude the thesis by discussing possible avenues for future research based on the methods and results present in the thesis.

Date of Award	22 May 2024
Original language	English
Awarding Institution	University of Bath
Supervisor	Roger Moser (Supervisor) & Hartmut Schwetlick (Supervisor)