Complete Geodesic Metrics in Big Classes
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911746851078144 |
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| author | Gupta, Prakhar |
| author_facet | Gupta, Prakhar |
| contents | Let $(X,ω)$ be a compact Kähler manifold and $θ$ 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 $\mathcal{E}^{p}(X,θ)$ can be endowed with a metric $d_{p}$ that makes $(\mathcal{E}^{p}(X,θ),d_{p})$ a complete geodesic metric space. The weak geodesics in $\mathcal{E}^{p}(X,θ)$ are the metric geodesic for $(\mathcal{E}^{p}(X,θ), d_{p})$. Moreover, for $p > 1$, the geodesic metric space $(\mathcal{E}^{p}(X,θ), d_{p})$ is uniformly convex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01688 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete Geodesic Metrics in Big Classes Gupta, Prakhar Differential Geometry Complex Variables 32U05 (Primary), 32Q15, 53C55 (Secondary) Let $(X,ω)$ be a compact Kähler manifold and $θ$ 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 $\mathcal{E}^{p}(X,θ)$ can be endowed with a metric $d_{p}$ that makes $(\mathcal{E}^{p}(X,θ),d_{p})$ a complete geodesic metric space. The weak geodesics in $\mathcal{E}^{p}(X,θ)$ are the metric geodesic for $(\mathcal{E}^{p}(X,θ), d_{p})$. Moreover, for $p > 1$, the geodesic metric space $(\mathcal{E}^{p}(X,θ), d_{p})$ is uniformly convex. |
| title | Complete Geodesic Metrics in Big Classes |
| topic | Differential Geometry Complex Variables 32U05 (Primary), 32Q15, 53C55 (Secondary) |
| url | https://arxiv.org/abs/2401.01688 |