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Isothermal coordinates

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In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form

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g = φ ( d x 1 2 + ⋯ + d x n 2 ) ,

where φ is a smooth function.

Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. On higher-dimensional Riemannian manifolds a necessary and sufficient condition for their local existence is the vanishing of the Weyl tensor and of the Cotton tensor.

Isothermal coordinates on surfaces

Gauss (1822) proved the existence of isothermal coordinates on an arbitrary surface with a real analytic metric, following results of Lagrange (1779) on surfaces of revolution. Results for Hölder continuous metrics were obtained by Korn (1916) and Lichtenstein (1916). Later accounts were given by Morrey (1938), Ahlfors (1955), Bers (1952) and Chern (1955). A particularly simple account using the Hodge star operator is given in DeTurck & Kazdan (1981).

Beltrami equation

The existence of isothermal coordinates can be proved by applying known existence theorems for the Beltrami equation, which rely on Lp estimates for singular integral operators of Calderón and Zygmund. A simpler approach to the Beltrami equation has been given more recently by the late Adrien Douady.

If the Riemannian metric is given locally as

d s 2 = E d x 2 + 2 F d x d y + G d y 2 ,

then in the complex coordinate z = x + iy, it takes the form

d s 2 = λ | d z + μ d z ¯ | 2 ,

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

λ = 1 4 ( E + G + 2 E G − F 2 ) , μ = ( E − G + 2 i F ) / 4 λ .

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

d s 2 = ρ ( d u 2 + d v 2 )

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

ρ | d w | 2 = ρ | w z | 2 | d z + w z ¯ w z d z ¯ | 2 ,

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

∂ w ∂ z ¯ = μ ∂ w ∂ z

has a diffeomorphic solution. Such a solution has been proved to exist in any neighbourhood where ||μ||∞ < 1.

Hodge star operator

New coordinates u and v are isothermal provided that

⋆ d u = d v ,

where ⋆ is the Hodge star operator defined by the metric.

Let Δ = 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.

By the Poincaré lemma ⋆ d u = d v has a local solution v exactly when d ⋆ d u = 0 .

Since

⋆ d ⋆ = 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.

Gaussian curvature

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

K = − 1 2 e − ρ ( ∂ 2 ρ ∂ u 2 + ∂ 2 ρ ∂ v 2 ) ,

where φ = e ρ .

References

Isothermal coordinates Wikipedia
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