The principle of least action in the space of Kähler potentials (Q2107728)

Given a compact Kähler manifold \((X,\omega)\), consider the associated Fréchet space of Kähler potentials \[ \mathcal H:=\{u\in\mathcal{C}^\infty(X)\mid \omega+dd^cu>0\}. \] It is endowed with a natural torsion-free metric connection \(\nabla\) introduced independently by Mabuchi and Semmes. The present paper studies the relation between geodesics on \(\mathcal H\) and parallel, fiberwise convex Lagrangians \[ L:T\mathcal H\cong\mathcal H\times\mathcal{C}^\infty(X)\rightarrow \mathbb{R}. \] More precisely, the author shows that, when \(L\) extends to the space of bounded \(\omega\)-plurisubharmonic functions and has a certain continuity property, then weak geodesics minimize the \(L\)-action \[ u\mapsto \int_0^TL(\dot{u}(t))dt. \] Under additional assumptions, also the converse holds, i.e. the minimizers of the action must be weak geodesics. Moreover, the author studies the variation of the least action associated to \(L\) along weak geodesics, proving certain convexity properties. These results generalize work of Calabi, Chen and Darvas among others, who treated the case where \(L\) is given by specific Finsler metrics.