Geodesics in the Space of m-Subharmonic Functions With Bounded Energy

Abstract

Abstract We raise our cups to Urban Cegrell, gone but not forgotten, gone but ever here. Until we meet again in Valhalla! With inspiration from the Kähler geometry, we introduce a metric structure on the energy class, $\mathcal {E}_{1,m}$, of $m$-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli–Nirenberg–Spruck model, we end by constructing geodesics in a subspace of our complete metric space.