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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.22667 |
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| _version_ | 1866909373093117952 |
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| author | Martin, Gaven Yao, Cong |
| author_facet | Martin, Gaven Yao, Cong |
| contents | We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:R \to S$ of finite distortion $\IK(z,f)$ between analytically finite Riemann surfaces in a fixed homotopy class $[f_0]$,\[ \mE_p(f:R,S)=\int_R \exp(p\IK(z,f))\; dσ(z). \] Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, $\mE_p(f_0,R,S)<\infty$. In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf differential, \[Φ=\exp(p\IK(w,h))\,h_w\overline{h_\wbar}\,dσ_R(h). \] From the Riemann-Roch theorem and an approximation technique, we show the variational equations hold for extremal mappings. We take this as a starting point for higher regularity to show that if $h:Ω\to\tildeΩ$ is a Sobolev homeomorphism between planar domains with holomorphic Ahlfors-Hopf differential, then $h$ is a diffeomorphism. It will follow that $h$ is harmonic in a metric induced by its own (smooth) distortion. We develop equations for the Beltrami coefficient of $h$, establishing a connection between degenerate elliptic non-linear Beltrami equations and these harmonic mappings. On the surface we conclude that minimisers $f_p\in [f_0]$ of $\mE_p(f:R,S)$ are diffeomorphisms and are unique stationary points. This now links two different approaches to Teichmüller theory; the classical theory of extremal quasiconformal maps and the harmonic mapping theory. As $p\to\infty$ we show $f_p\to f_\infty$ to recover the unique extremal quasiconformal mapping . This extremal quasiconformal mapping is not a diffeomorphism (unless it is conformal) and $f_p$ degenerates on a divisor. As $p\to0$ we recover the harmonic diffeomorphism in $[f_0]$ and Shoen-Yau's results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The exponential Teichmüller theory: Ahlfors--Hopf differentials and diffeomorphisms Martin, Gaven Yao, Cong Complex Variables 30C We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:R \to S$ of finite distortion $\IK(z,f)$ between analytically finite Riemann surfaces in a fixed homotopy class $[f_0]$,\[ \mE_p(f:R,S)=\int_R \exp(p\IK(z,f))\; dσ(z). \] Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, $\mE_p(f_0,R,S)<\infty$. In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf differential, \[Φ=\exp(p\IK(w,h))\,h_w\overline{h_\wbar}\,dσ_R(h). \] From the Riemann-Roch theorem and an approximation technique, we show the variational equations hold for extremal mappings. We take this as a starting point for higher regularity to show that if $h:Ω\to\tildeΩ$ is a Sobolev homeomorphism between planar domains with holomorphic Ahlfors-Hopf differential, then $h$ is a diffeomorphism. It will follow that $h$ is harmonic in a metric induced by its own (smooth) distortion. We develop equations for the Beltrami coefficient of $h$, establishing a connection between degenerate elliptic non-linear Beltrami equations and these harmonic mappings. On the surface we conclude that minimisers $f_p\in [f_0]$ of $\mE_p(f:R,S)$ are diffeomorphisms and are unique stationary points. This now links two different approaches to Teichmüller theory; the classical theory of extremal quasiconformal maps and the harmonic mapping theory. As $p\to\infty$ we show $f_p\to f_\infty$ to recover the unique extremal quasiconformal mapping . This extremal quasiconformal mapping is not a diffeomorphism (unless it is conformal) and $f_p$ degenerates on a divisor. As $p\to0$ we recover the harmonic diffeomorphism in $[f_0]$ and Shoen-Yau's results. |
| title | The exponential Teichmüller theory: Ahlfors--Hopf differentials and diffeomorphisms |
| topic | Complex Variables 30C |
| url | https://arxiv.org/abs/2410.22667 |