Finite Energy Geodesic Rays in Big Cohomology Classes
Abstract
For a big class represented by $\theta$, we show that the metric space $(\mathcal{E}^{p}(X,\theta),d_{p})$ for $p \geq 1$ is Buseman convex. This allows us to construct a chordal metric $d_{p}^{c}$ on the space of geodesic rays in $\mathcal{E}^{p}(X,\theta)$. We also prove that the space of finite $p$energy geodesic rays with the chordal metric $d_{p}^{c}$ is a complete geodesic metric space. With the help of the metric $d_{p}$, we find a characterization of geodesic rays lying in $\mathcal{E}^{p}(X,\theta)$ in terms of the corresponding test curves via the RossWitt Nyström correspondence. This result is new even in the Kähler setting.
 Publication:

arXiv eprints
 Pub Date:
 June 2024
 DOI:
 10.48550/arXiv.2406.07669
 arXiv:
 arXiv:2406.07669
 Bibcode:
 2024arXiv240607669G
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Complex Variables;
 Primary: 32U05;
 Secondary: 32Q15;
 53C55
 EPrint:
 19 pages. Comments are Welcome