Schwarz-Pick lemma for harmonic maps which are conformal at a point
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910427475083264 |
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| author | Forstneric, Franc Kalaj, David |
| author_facet | Forstneric, Franc Kalaj, David |
| contents | We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb D\to \mathbb B^n$. This implies that conformal harmonic immersions $M \to \mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$\mathrm{é}$ metric on $M$ and the Cayley-Klein metric on the ball $\mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $\mathbb D$ onto affine discs in $\mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $\mathbb R^n$ based on minimal surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_12403 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Schwarz-Pick lemma for harmonic maps which are conformal at a point Forstneric, Franc Kalaj, David Differential Geometry Complex Variables Primary: 53A10, Secondary: 30C80, 31A05, 32Q45 We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $\mathbb D$ in $\mathbb C$ into the unit ball $\mathbb B^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb D\to \mathbb B^n$. This implies that conformal harmonic immersions $M \to \mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$\mathrm{é}$ metric on $M$ and the Cayley-Klein metric on the ball $\mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $\mathbb D$ onto affine discs in $\mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $\mathbb R^n$ based on minimal surfaces. |
| title | Schwarz-Pick lemma for harmonic maps which are conformal at a point |
| topic | Differential Geometry Complex Variables Primary: 53A10, Secondary: 30C80, 31A05, 32Q45 |
| url | https://arxiv.org/abs/2102.12403 |