Complete geodesic metrics in big classes

Source abstract

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