Complete Geodesic Metrics in Big Classes

Abstract

Let $(X,\omega)$ be a compact K\"ahler 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.