Complete Geodesic Metrics in Big Classes

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 $p\ge 1$, 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.