Complete geodesic metrics in big classes

Abstract:

Let $(X,\omega )$ be a compact Kähler manifold and $\theta$ be a smooth closed real $(1,1)$-form that represents a big cohomology class. In this paper, we show that for $p\geq 1$, the high energy space $\mathcal {E}^{p}(X,\theta )$ can be endowed with a metric $d_{p}$ that makes $(\mathcal {E}^{p}(X,\theta ),d_{p})$ a complete geodesic metric space. The weak geodesics in $\mathcal {E}^{p}(X,\theta )$ are the metric geodesic for $(\mathcal {E}^{p}(X,\theta ), d_{p})$. Moreover, for $p > 1$, the geodesic metric space $(\mathcal {E}^{p}(X,\theta ), d_{p})$ is uniformly convex.