Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric.
One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane.
Gauss proved the existence of isothermal coordinates locally in the analytic case by reducing the Beltrami to an ordinary differential equation in the complex domain.
An isothermal coordinate system, say in a neighborhood of the origin (x, y) = (0, 0), is given by the real and imaginary parts of a complex-valued function f(x, y) that satisfies Let
Gauss lets q(t) be some complex-valued function of a real variable t that satisfies the following ordinary differential equation: where E, F, and G are here evaluated at y = t and x = q(t).
, the vector (dx, dy) is then tangent to the solution curve of the differential equation that passes through the point (x, y).
In the simplest cases the Beltrami equation can be solved using only Hilbert space techniques and the Fourier transform.
Now in the Beltrami equation with μ a smooth function of compact support, set and assume that the first derivatives of g are L2.
In fact formally writing fz = ek, it follows that This equation for k can be solved by the same methods as above giving a solution tending to 0 at ∞.
Conversely Beltrami coefficients defined on the closures of the upper halfplane or unit disk which satisfy these conditions on the boundary can be "reflected" using the formulas above.
From the composition formulas for Beltrami coefficients so that f = F∘ h−1 is a smooth diffeomorphism between the closures of D and U which is holomorphic on the interior.
Let After a change of variable in the t coordinate and a conformal change in the metric, the metric takes the form where ψ is an analytic real-valued function of t: A formal diffeomorphism sending (θ,t) to (f(θ,t),t) can be defined as a formal power series in t: where the coefficients fn are smooth functions on the circle.
This condition is imposed by demanding that no odd powers of t appear in the formal power series expansion: By Borel's lemma, there is a diffeomorphism defined in a neighbourhood of the unit circle, t = 0, for which the formal expression f(θ,t) is the Taylor series expansion in the t variable.
Douady and others have indicated ways to extend the L2 theory to prove the existence and uniqueness of solutions when the Beltrami coefficient μ is bounded and measurable with L∞ norm k strictly less than one.
Their approach involved the theory of quasiconformal mappings to establish directly the solutions of Beltrami's equation when μ is smooth with fixed compact support are uniformly Hölder continuous.
In the Beltrami equation with μ a smooth function of compact support, set and assume that the first derivatives of g are Lp.
The corresponding normalised solutions fn of the Beltrami equations and their inverses gn satisfy uniform Hölder estimates.
They are therefore equicontinuous on any compact subset of C; they are even holomorphic for |z| > R. So by the Arzelà–Ascoli theorem, passing to a subsequence if necessary, it can be assumed that both fn and gn converge uniformly on compacta to f and g. The limits will satisfy the same Hölder estimates and be holomorphic for |z| > R. The relations fn∘gn = id = gn∘fn imply that in the limit f∘g = id = g∘f, so that f and g are homeomorphisms.
Hence and so differentiating If g is another solution then Since Tμ has operator norm on Lp less than 1, this forces But then from the Beltrami equation Hence f − g is both holomorphic and antiholomorphic, so a constant.
Since f(0) = 0 = g(0), it follows that f = g. Note that since f is holomorphic off the support of μ and f(∞) = ∞, the conditions that the derivatives are locally in Lp force For a general f satisfying Beltrami's equation and with distributional derivatives locally in Lp, it can be assumed after applying a Möbius transformation that 0 is not in the singular set of the Beltrami coefficient μ.
The images of the fundamental domain under the group fill out C with 0 removed and the Beltrami coefficient is smooth there.
For regions with a higher degree of connectivity k + 1, the result is essentially Bers' generalization of the retrosection theorem.
The interior of Ω1 iz a fundamental domain for G. Moreover, the index two normal subgroup G0 consisting of orientation-preserving mappings is a classical Schottky group.
The normalised solution of the Beltrami equation h is a smooth diffeomorphism of the closure of Ω1 onto itself preserving the unit circle, its exterior and interior.
As topological spaces M1 and M2 are homeomorphic to a fixed quotient of the upper half plane H by a discrete cocompact subgroup Γ of PSL(2,R).
A result of Munkres (1960) implies that the homeomorphisms can be adjusted near the edges and the vertices of the triangulation to produce diffeomorphisms.
Let μ be the corresponding Beltrami coefficient on H. It can be extended to C by reflection It satisfies the invariance property for g in Γ.
The solution f of the corresponding Beltrami equation defines a homeomorphism of C, preserving the real axis and the upper and lower half planes.
It defines a Beltrami coefficient λ omn H which this time is extended to C by defining λ to be 0 off H. The solution h of the Beltrami equation is a homeomorphism of C which is holomorphic on the lower half plane and smooth on the upper half plane.
Douady & Earle (1986), generalizing earlier results of Ahlfors and Beurling, produced such an extension with the additional properties that it commutes with the action of SU(1,1) by Möbius transformations and is quasiconformal if f is quasisymmetric.