Hyperbolic domains in real Euclidean spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866913331623755776 |
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| author | Drnovsek, Barbara Drinovec Forstneric, Franc |
| author_facet | Drnovsek, Barbara Drinovec Forstneric, Franc |
| contents | The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb R^n$, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_06943 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hyperbolic domains in real Euclidean spaces Drnovsek, Barbara Drinovec Forstneric, Franc Complex Variables Differential Geometry Primary 53A10. Secondary 32Q45, 30C80, 31A05 The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb R^n$, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. |
| title | Hyperbolic domains in real Euclidean spaces |
| topic | Complex Variables Differential Geometry Primary 53A10. Secondary 32Q45, 30C80, 31A05 |
| url | https://arxiv.org/abs/2109.06943 |