Isothermal coordinates

The existence of isothermal coordinates can be proved[1] by applying known existence theorems for the Beltrami equation, which rely on L^p estimates for singular integral operators of Calderón and Zygmund.[2][3] A simpler approach to the Beltrami equation has been given more recently by Adrien Douady.[4]

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 z = x + iy, it takes the form

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

where λ and μ are smooth with λ > 0 and |μ| < 1. In fact

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

In isothermal coordinates (u, v) the metric should take the form

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

with ρ > 0 smooth. The complex coordinate w = u + i v satisfies

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

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 ||μ||_∞ < 1.

New coordinates u and v are isothermal provided that

${\displaystyle \star du=dv,}$

where ${\displaystyle \star }$ is the Hodge star operator defined by the metric.[5]

Let ${\displaystyle \Delta =d^{*}d}$ be the Laplace–Beltrami operator on functions.

Then by standard elliptic theory, u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du non-vanishing.[5][6]

Indeed, since the problem is local, it is sufficient to describe a solution on the torus T² endowed with a Riemannian metric. In this case Δ f = g can be solved near 0 with given initial values f(0), df(0).

This can be proved using the L² Sobolev spaces H_s(T²) for s ≥ 0.[7] These Hilbert spaces can be defined in terms of Δ and the Riemannian structure, but are independent of these structures. It follows that I + Δ gives a linear isomorphism from H_s+2(T²) onto H_s(T²) and that Δ f = g is solvable if and only if g is orthogonal to the constants. On the other hand standard techniques imply an approximation theorem:[8] smooth functions which vanish in a neighbourhood of a point are dense in H_s(T²) for s ≤ 1 (for the method of proof, see below).

In particular density implies that for any s > 0 small there are smooth functions g equal to 0 near 1, orthogonal to the constants in H_s(T²) such that the functions f = ∆⁻¹ g are dense in the subspace of H_s+2(T²) orthogonal to constants. By elliptic regularity, these f are smooth. By the Sobolev embedding theorem H_s+2(T²) lies in C¹(T²); density in the Sobolev space implies that f(0), df(0) take all possible values, as claimed.

The approximation theorem above can be proved using the same methods as the corresponding 1-dimensional result: smooth functions which vanish in a neighbourhood of a point are dense in H_s(T) for s ≤ 1/2. For simplicity only this case will be described. It suffices to prove this for the point 1 on the unit circle T. By Cayley transform between the circle and the real line, functions vanishing to infinite order at 1 in C^∞(T) can be identified with S(R), the space of Schwartz functions on R. Smooth functions of compact support are dense in S(R); and hence A the space of smooth functions vanishing in a neighbourhood of 1 in C^∞(T) is dense in the space of smooth functions vanishing with all their derivatives at 1. By the Stone-Weierstrass theorem, A is uniformly dense in C₀(T\{1}). Thus if h lies in B, the ideal in C¹(T) of functions vanishing with their derivative at 1, h and h' can be uniformly approximated by a function in A. Hence A is dense in B. On the other hand C¹(T) is in H_s(T) if s ≤ 1/2. To prove that A is dense in H_s(T), it therefore suffices to show it contains functions a_n(θ}}) and b_n(θ) tending to zero in Sobolev norm with a_n(0) = 0 vanishing at 1 and ∂_θa_n(0) = 1; and b_n(0) = 1 ad ∂_θb_n(0) = 0. Suitable functions are a_n(θ) = sin nθ / n and b_n(θ) = c_n(θ) / c_n(0) where c_n(θ) = ∑ (1 − n⁻¹)^k cos kθ / k log k}}.[9]

By the Poincaré lemma ${\displaystyle \star du=dv}$ has a local solution v exactly when ${\displaystyle d\star du=0}$.

Since

${\displaystyle \star d\star =d^{*},}$

this is equivalent to Δ u = 0, and hence a local solution exists.

Since du is non-zero and the square of the Hodge star operator is −1 on 1-forms, du and dv are necessarily linearly independent, and therefore u and v give local isothermal coordinates.

In the isothermal coordinates (u, v), the Gaussian curvature takes the simpler form

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

where ${\displaystyle \varphi =e^{\rho }}$.