Isothermal coordinates

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

${\displaystyle g=\varphi (dx_{1}^{2}+\cdots +dx_{n}^{2}),}$

where ${\displaystyle \varphi }$ is a smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)

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) harvtxt error: no target: CITEREFAhlfors1955 (help), 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[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.

Hodge star operator

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.

Gaussian curvature

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

See also

• Conformal map
• Liouville's equation
• Quasiconformal map

Notes

1. Imayoshi & Taniguchi 1992, pp. 20–21
2. Ahlfors 1966, pp. 85–115
3. Imayoshi & Taniguchi 1992, pp. 92–104
4. Douady & Buff 2000
5. DeTurck & Kazdan 1981; Taylor 1996, pp. 377–378
6. For an alternative proof using potential theory and singular integral operators, see Bers, John & Schechter 1979, pp. 228–230
7. See:
   • Bers, John & Schechter 1979
   • Warner 1983
   • Taylor 1996
8. Hörmander 1990
9. Zygmund 2002 harvnb error: no target: CITEREFZygmund2002 (help)

References

• Ahlfors, Lars V. (1952), Conformality with respect to Riemannian metrics., Ann. Acad. Sci. Fenn. Ser. A. I., 206, pp. 1–22
• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Bers, Lipman (1952), Riemann Surfaces, 1951–1952, New York University, pp. 15–35
• Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, ISBN 0-8218-0049-3
• Chern, Shiing-shen (1955), "An elementary proof of the existence of isothermal parameters on a surface", Proc. Amer. Math. Soc., American Mathematical Society, 6 (5): 771–782, doi:10.2307/2032933, JSTOR 2032933
• DeTurck, Dennis M.; Kazdan, Jerry L. (1981), "Some regularity theorems in Riemannian geometry", Annales Scientifiques de l'École Normale Supérieure, Série 4, 14 (3): 249–260, doi:10.24033/asens.1405, ISSN 0012-9593, MR 0644518.
• do Carmo, Manfredo (1976), Differential Geometry of Curves and Surfaces, Prentice Hall, ISBN 0-13-212589-7
• Douady, Adrien; Buff, X. (2000), Le théorème d'intégrabilité des structures presque complexes. [Integrability theorem for almost complex structures], London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, pp. 307–324
• Gauss, C.F. (1822), On Conformal Representation, translator Smith, Eugene, pp. 463–475
• Hörmander, Lars (1990), The analysis of linear partial differential operators I,Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256 (Second ed.), Springer-Verlag, ISBN 3-540-52345-6
• Imayoshi, Y.; Taniguchi, M. (1992), An Introduction to Teichmüller spaces, Springer-Verlag, ISBN 0-387-70088-9
• Korn, A. (1916), Zwei Anwendungen der Methode der sukzessiven Annäherungen, Schwarz Abhandlungen, pp. 215–219
• Lagrange, J. (1779), Sur la construction des cartes géographiques
• Lichtenstein, L. (1916), "Zur Theorie der konformen Abbildung", Bull. Internat. Acad. Sci. Cracovie. Cl. Sci. Math. Nat. Sér. A.: 192–217
• Morrey, Charles B. (1938), "On the solutions of quasi-linear elliptic partial differential equations", Trans. Amer. Math. Soc., American Mathematical Society, 43 (1): 126–166, doi:10.2307/1989904, JSTOR 1989904
• Spivak, Michael, A Comprehensive Introduction to Differential Geometry, 4 (3 ed.), Publish or Perish, pp. 314–346
• Taylor, Michael E. (1996), Partial Differential Equations: Basic Theory, Springer-Verlag, pp. 376–378, ISBN 0-387-94654-3
• Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer-Verlag, ISBN 0-387-90894-3

External links

• "Isothermal coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]