Hyperbolic curvature of holomorphic level curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918387563626496 |
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| author | Iancu, Mihai Nechita, Veronica-Oana |
| author_facet | Iancu, Mihai Nechita, Veronica-Oana |
| contents | We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence, we point out a rigidity theorem: if the hyperbolic curvature of the above level curve vanishes at some point, then the level curve is a hyperbolic geodesic and $f$ is an automorphism. As another consequence, we prove that $\frac{1}{\sqrt 2}$ is the greatest lower bound of the supremum of $r\in(0,1)$ such that the level curve $|z|=r|f(z)|$ is (Euclidean) convex. This constant turns out to be also the radius of convexity for hyperbolically convex self-maps of $\mathbb{D}$ that fix the origin. We also give (sharp) estimates for the total hyperbolic curvature, hyperbolic area and hyperbolic perimeter of the sublevel sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperbolic curvature of holomorphic level curves Iancu, Mihai Nechita, Veronica-Oana Complex Variables 30F45, 30C45, 30C35, 52A10 We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence, we point out a rigidity theorem: if the hyperbolic curvature of the above level curve vanishes at some point, then the level curve is a hyperbolic geodesic and $f$ is an automorphism. As another consequence, we prove that $\frac{1}{\sqrt 2}$ is the greatest lower bound of the supremum of $r\in(0,1)$ such that the level curve $|z|=r|f(z)|$ is (Euclidean) convex. This constant turns out to be also the radius of convexity for hyperbolically convex self-maps of $\mathbb{D}$ that fix the origin. We also give (sharp) estimates for the total hyperbolic curvature, hyperbolic area and hyperbolic perimeter of the sublevel sets. |
| title | Hyperbolic curvature of holomorphic level curves |
| topic | Complex Variables 30F45, 30C45, 30C35, 52A10 |
| url | https://arxiv.org/abs/2511.00227 |