Two optimization problems for the Loewner energy
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915118084784128 |
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| author | Wang, Yilin |
| author_facet | Wang, Yilin |
| contents | A Jordan curve on the Riemann sphere can be encoded by its conformal welding, a circle homeomorphism. The Loewner energy measures how far a Jordan curve is away from being a circle, or equivalently, how far its welding homeomorphism is away from being a Möbius transformation. We consider two optimizing problems for the Loewner energy, one under the constraint for the curves to pass through $n$ given points on the Riemann sphere, which is the conformal boundary of hyperbolic $3$-space $\mathbb H^3$; the other under the constraint for $n$ given points on the circle to be welded to another $n$ given points of the circle. The latter problem can be viewed as optimizing positive curves on the boundary of AdS$^3$ space passing through $n$ prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in $\partial_\infty \mathbb H^3$ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS$^3$, whereas optimizing the positive curve in $\partial_\infty\!\operatorname{AdS}^3$ gives rise to a Jordan curve that is the boundary of a pleated plane in $\mathbb H^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two optimization problems for the Loewner energy Wang, Yilin Complex Variables Differential Geometry Geometric Topology Probability A Jordan curve on the Riemann sphere can be encoded by its conformal welding, a circle homeomorphism. The Loewner energy measures how far a Jordan curve is away from being a circle, or equivalently, how far its welding homeomorphism is away from being a Möbius transformation. We consider two optimizing problems for the Loewner energy, one under the constraint for the curves to pass through $n$ given points on the Riemann sphere, which is the conformal boundary of hyperbolic $3$-space $\mathbb H^3$; the other under the constraint for $n$ given points on the circle to be welded to another $n$ given points of the circle. The latter problem can be viewed as optimizing positive curves on the boundary of AdS$^3$ space passing through $n$ prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in $\partial_\infty \mathbb H^3$ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS$^3$, whereas optimizing the positive curve in $\partial_\infty\!\operatorname{AdS}^3$ gives rise to a Jordan curve that is the boundary of a pleated plane in $\mathbb H^3$. |
| title | Two optimization problems for the Loewner energy |
| topic | Complex Variables Differential Geometry Geometric Topology Probability |
| url | https://arxiv.org/abs/2402.10054 |