Towards a Theory of Multi-Parameter Geometrical Variational Problems: Fibre Bundles, Differential Forms and Riemannian Quasiconvexity

Abstract: We are concerned with the relaxation and existence theories of a general class of geometrical minimisation problems, with action integrals defined via differential forms over fibre bundles. We find natural algebraic and analytic conditions which give rise to a relaxation theory. Moreover, we propose the notion of ``Riemannian quasiconvexity'' for cost functions whose variables are differential forms on Riemannian manifolds, which extends the classical quasiconvexity condition in the Euclidean settings. The existence of minimisers under the Riemannian quasiconvexity condition has been established. This work may serve as a tentative generalisation of the framework developed in the paper: B. Dacorogna and W. Gangbo, Quasiconvexity and relaxation in optimal transportation of closed differential forms, \textit{Arch. Ration. Mech. Anal.} (2019), to appear. DOI: \texttt{https://doi.org/10.1007/s00205-019-01390-9}.