Beltrami equation

The existence of isothermal coordinates can be proved^[6] by applying known existence theorems for the Beltrami equation, which rely on L^p estimates for singular integral operators of Calderón and Zygmund.^[7][8] A simpler approach to the Beltrami equation has been given more recently by Adrien Douady.^[9]

If the Riemannian metric is given locally as

${\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}$

then in the complex coordinate ${\displaystyle z=x+iy}$, it takes the form

${\displaystyle ds^{2}=\lambda |\,dz+\mu \,d{\overline {z}}|^{2},}$

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are smooth with ${\displaystyle \lambda >0}$ and ${\displaystyle \left\vert \mu \right\vert <1}$. In fact

${\displaystyle \lambda ={1 \over 4}(E+G+2{\sqrt {EG-F^{2}}}),\,\,\,{\displaystyle \mu ={(E-G+2iF) \over 4\lambda }}.}$

In isothermal coordinates ${\displaystyle (u,v)}$ the metric should take the form

${\displaystyle ds^{2}=e^{\rho }(du^{2}+dv^{2})}$

with ρ smooth. The complex coordinate ${\displaystyle w=u+iv}$ satisfies

${\displaystyle e^{\rho }\,|dw|^{2}=e^{\rho }|w_{z}|^{2}|\,dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}|^{2},}$

so that the coordinates (u, v) will be isothermal if the Beltrami equation

${\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}$

has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where ${\displaystyle \lVert \mu \rVert _{\infty }<1}$.

Existence via local solvability for elliptic partial differential equations

The existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations. In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to the Riemannian metric. The local solvability then states that any point p has a neighborhood U on which there is a harmonic function u with nowhere-vanishing derivative.^[10]

Isothermal coordinates are constructed from such a function in the following way.^[11] Harmonicity of u is identical to the closedness of the differential 1-form ${\displaystyle \star du,}$ defined using the Hodge star operator ${\displaystyle \star }$ associated to the Riemannian metric. The Poincaré lemma thus implies the existence of a function v on U with ${\displaystyle dv=\star du.}$ By definition of the Hodge star, ${\displaystyle du}$ and ${\displaystyle dv}$ are orthogonal to one another and hence linearly independent, and it then follows from the inverse function theorem that u and v form a coordinate system on some neighborhood of p. This coordinate system is automatically isothermal, since the orthogonality of ${\displaystyle du}$ and ${\displaystyle dv}$ implies the diagonality of the metric, and the norm-preserving property of the Hodge star implies the equality of the two diagonal components.

Gaussian curvature

In the isothermal coordinates ${\displaystyle (u,v)}$, the Gaussian curvature takes the simpler form

${\displaystyle K=-{\frac {1}{2}}e^{-\rho }\left({\frac {\partial ^{2}\rho }{\partial u^{2}}}+{\frac {\partial ^{2}\rho }{\partial v^{2}}}\right).}$