In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation
for a complex distribution of the complex variable in some open set , with derivatives that are locally L2, and where is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient, and where and are Wirtinger derivatives. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the Lp theory of the Beurling transform, a singular integral operator defined on Lp(C) for all 1 < p < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain.
Metrics on planar domains
Consider a 2-dimensional Riemannian manifold, say with an (x, y) coordinate system on it. The curves of constant x on that surface typically don't intersect the curves of constant y orthogonally. A new coordinate system (u, v) is called isothermal when the curves of constant u do intersect the curves of constant v orthogonally and, in addition, the parameter spacing is the same — that is, for small enough h, the little region with and is nearly square, not just nearly rectangular. The Beltrami equation is the equation that has to be solved in order to construct isothermal coordinate systems.
To see how this works, let S be an open set in C and let
be a smooth metric g on S. The first fundamental form of g
is a positive real matrix (E > 0, G > 0, EG − F2 > 0) that varies smoothly with x and y.
The Beltrami coefficient of the metric g is defined to be
This coefficient has modulus strictly less than one since the identity
implies that
Let f(x,y) =(u(x,y),v(x,y)) be a smooth diffeomorphism of S onto another open set T in C. The map f preserves orientation just when its Jacobian is positive:
And using f to pull back to S the standard Euclidean metric ds2 = du2 + dv2 on T induces a metric on S given by
a metric whose first fundamental form is
When f both preserves orientation and induces a metric that differs from the original metric g only by a positive, smoothly varying scale factor r(x, y), the new coordinates u and v defined on S by f are called isothermal coordinates.
To determine when this happens, we reinterpret f as a complex-valued function of a complex variable f(x+iy) = u(x+iy) + iv(x+iy) so that we can apply the Wirtinger derivatives:
Since
the metric induced by f is given by
The Beltrami quotient of this induced metric is defined to be .
The Beltrami quotient of equals the Beltrami coefficient of the original metric g just when
The real and imaginary parts of this identity linearly relate and and solving for and gives
It follows that the metric induced by f is then r(x, y) g(x,y), where which is positive, while the Jacobian of f is then which is also positive. So, when the new coordinate system given by f is isothermal.
Conversely, consider a diffeomorphism f that does give us isothermal coordinates. We then have
where the scale factor r(x, y) has dropped out and the expression inside the square root is the perfect square Since f must preserve orientation to give isothermal coordinates, the Jacobian is the positive square root; so we have
The right-hand factors in the numerator and denominator are equal and, since the Jacobian is positive, their common value can't be zero; so
Thus, the local coordinate system given by a diffeomorphism f is isothermal just when f solves the Beltrami equation for
Isothermal coordinates for analytic metrics
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. Here is a cookbook presentation of Gauss's technique.
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 be such a function, and let be a complex-valued function of a complex variable that is holomorphic and whose derivative is nowhere zero. Since any holomorphic function has identically zero, we have
Thus, the coordinate system given by the real and imaginary parts of is also isothermal. Indeed, if we fix to give one isothermal coordinate system, then all of the possible isothermal coordinate systems are given by for the various holomorphic with nonzero derivative.
When E, F, and G are real analytic, Gauss constructed a particular isothermal coordinate system the one that he chose being the one with for all x. So the u axis of his isothermal coordinate system coincides with the x axis of the original coordinates and is parameterized in the same way. All other isothermal coordinate systems are then of the form for a holomorphic with nonzero derivative.
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). If we specify the value of q(s) for some start value s, this differential equation determines the values of q(t) for t either less than or greater than s. Gauss then defines his isothermal coordinate system h by setting h(x, y) to be along the solution path of that differential equation that passes through the point (x, y), and thus has q(y) = x.
This rule sets h(x, 0) to be , since the starting condition is then q(0)=x. More generally, suppose that we move by an infinitesimal vector (dx, dy) away from some point (x, y), where dx and dy satisfy
Since , the vector (dx, dy) is then tangent to the solution curve of the differential equation that passes through the point (x, y). Because we are assuming the metric to be analytic, it follows that
for some smooth, complex-valued function We thus have
We form the quotient and then multiply numerator and denominator by , which is the complex conjugate of the denominator. Simplifying the result, we find that
Gauss's function h thus gives the desired isothermal coordinates.
Solution in L2 for smooth Beltrami coefficients In the simplest cases the Beltrami equation can be solved using only Hilbert space techniques and the Fourier transform. The method of proof is the prototype for the general solution using Lp spaces, although Adrien Douady has indicated a method for handling the general case using only Hilbert spaces: the method relies on the classical theory of quasiconformal mappings to establish Hölder estimates that are automatic in the Lp theory for p > 2.See: Let T be the Beurling transform on L2(C) defined on the Fourier transform of an L2 function f as a multiplication operator:
It is a unitary operator and if f is a tempered distribution on C with partial derivatives in L2 then
where the subscripts denote complex partial derivatives.
The fundamental solution of the operator
is given by the distribution
a locally integrable function on C. Thus on Schwartz functions f
The same holds for distributions of compact support on C. In particular if f is an L2 function with compact support, then its Cauchy transform, defined as
is locally square integrable. The above equation can be written
Moreover, still regarding f and Cf as distributions,
Indeed, the operator D is given on Fourier transforms as multiplication by iz/2 and C as multiplication by its inverse.
Now in the Beltrami equation
with μ a smooth function of compact support, set
and assume that the first derivatives of g are L2. Let h = gz = fz – 1. Then
If A and B are the operators defined by
then their operator norms are strictly less than 1 and
Hence
where the right hand sides can be expanded as Neumann series. It follows that
has the same support as μ and g. Hence f is given by
Elliptic regularity can now be used to deduce that f is smooth.
In fact, off the support of μ,
so by Weyl's lemma f is even holomorphic for |z| > R. Since f = CT*h + z, it follows that f tends to 0 uniformly as |z| tends to ∞.
The elliptic regularity argument to prove smoothness, however, is the same everywhere and uses the theory of L2 Sobolev spaces on the torus.See: Let ψ be a smooth function of compact support on C, identically equal to 1 on a neighbourhood of the support of μ and set F = ψ f. The support of F lies in a large square |x|, |y| ≤ R, so, identifying opposite sides of the square, F and μ can be regarded as a distribution and smooth function on a torus T2. By construction F is in L2(T2). As a distribution on T2 it satisfies
where G is smooth. On the canonical basis em of L2(T2) with m in Z + i Z, define
Thus U is a unitary and on trigonometric polynomials or smooth functions P
Similarly it extends to a unitary on each Sobolev space Hk(T2) with the same property. It is the counterpart on the torus of the Beurling transform. The standard theory of Fredholm operators shows that the operators corresponding to I – μ U and I – U μ are invertible on each Sobolev space. On the other hand,
Since UG is smooth, so too is (I – μU)F and hence also F.
Thus the original function f is smooth. Regarded as a map of C = R2 into itself, the Jacobian is given by
This Jacobian is nowhere vanishing by a classical argument of . 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 ∞. By uniqueness h + 1 = ek so that
is nowhere vanishing. Since f induces a smooth map of the Riemann sphere C ∪ ∞ into itself which is locally a diffeomorphism, f must be a diffeomorphism. In fact f must be onto by connectedness of the sphere, since its image is an open and closed subset; but then, as a covering map, f must cover each point of the sphere the same number of times. Since only ∞ is sent to ∞, it follows that f is one-to-one.
The solution f is a quasiconformal conformal diffeomorphism. These form a group and their Beltrami coefficients can be computed according to the following rule:See:
Moreover, if f(0) = 0 and
then
This formula reflects the fact that on a Riemann surface, a Beltrami coefficient is not a function. Under a holomorphic change of coordinate w = w(z), the coefficient is transformed to
Defining a smooth Beltrami coefficient on the sphere in this way, if μ is such a coefficient then, taking a smooth bump function ψ equal to 0 near 0, equal 1 for |z| > 1 and satisfying 0 ≤ ψ ≤ 1, μ can be written as a sum of two Beltrami coefficients:
Let g be the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient μ∞. Let λ be the Beltrami coefficient of compact support on C defined by
If f is the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient λ, then the transformation formulas above show that f g−1 is a quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient μ.
The solutions of Beltrami's equation restrict to diffeomorphisms of the upper halfplane or unit disk if the coefficient μ has extra symmetry properties;See since the two regions are related by a Möbius transformation (the Cayley transform), the two cases are essentially the same.