The Loewner energy of loops and regularity of driving functions
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866914664684716032 |
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| author | Rohde, Steffen Wang, Yilin |
| author_facet | Rohde, Steffen Wang, Yilin |
| contents | Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,β}$ curve (differentiable parametrization with $β$-Hölder continuous derivative) is in the class $C^{1,β-1/2}$ if $1/2<β\leq 1$, and in the class $C^{0,β+ 1/2}$ if $0 \leq β\leq 1/2$. This is the converse of a result of Carto Wong and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_04959 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The Loewner energy of loops and regularity of driving functions Rohde, Steffen Wang, Yilin Complex Variables Probability Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,β}$ curve (differentiable parametrization with $β$-Hölder continuous derivative) is in the class $C^{1,β-1/2}$ if $1/2<β\leq 1$, and in the class $C^{0,β+ 1/2}$ if $0 \leq β\leq 1/2$. This is the converse of a result of Carto Wong and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. |
| title | The Loewner energy of loops and regularity of driving functions |
| topic | Complex Variables Probability |
| url | https://arxiv.org/abs/1710.04959 |