Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited

We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note “Some useful techniques for dealing with multiple valued functions” and generalizes Almgren’s $Q$-valued functions. We study some relevant properties of such $Q$-multisections and apply the theory to provide an elementary and purely geometric proof of a delicate reparametrization theorem for multi-valued graphs which plays an important role in the regularity theory for higher codimension area minimizing currents à la Almgren—De Lellis—Spadaro.

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Received 27 May 2017

Accepted 5 August 2019

Published 22 July 2022