Finite Energy Geodesic Rays in Big Cohomology Classes
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909222115999744 |
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| author | Gupta, Prakhar |
| author_facet | Gupta, Prakhar |
| contents | For a big class represented by $θ$, we show that the metric space $(\mathcal{E}^{p}(X,θ),d_{p})$ 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 $\mathcal{E}^{p}(X,θ)$. 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 $\mathcal{E}^{p}(X,θ)$ in terms of the corresponding test curves via the Ross-Witt Nyström correspondence. This result is new even in the Kähler setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_07669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite Energy Geodesic Rays in Big Cohomology Classes Gupta, Prakhar Differential Geometry Complex Variables Primary: 32U05, Secondary: 32Q15, 53C55 For a big class represented by $θ$, we show that the metric space $(\mathcal{E}^{p}(X,θ),d_{p})$ 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 $\mathcal{E}^{p}(X,θ)$. 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 $\mathcal{E}^{p}(X,θ)$ in terms of the corresponding test curves via the Ross-Witt Nyström correspondence. This result is new even in the Kähler setting. |
| title | Finite Energy Geodesic Rays in Big Cohomology Classes |
| topic | Differential Geometry Complex Variables Primary: 32U05, Secondary: 32Q15, 53C55 |
| url | https://arxiv.org/abs/2406.07669 |