This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities.
These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann.
Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces.
These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces.
This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.
On a Riemann surface the Hodge star is defined on 1-forms by the local formula It is well-defined because it is invariant under holomorphic changes of coordinate.
They have the properties On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact.
Note that if ω has compact support, so vanishes outside some smaller rectangle (a1,b1) × (c1,d1) with a < a1 < b1
In local coordinates if ω = p dx + q dy and γ(t) = (x(t),y(t)) then so that Note that if the 1-form ω is exact on some connected open set U, so that ω = df for some smooth function f on U (unique up to a constant), and γ(t), a ≤ t ≤ b, is a smooth path in U, then This depends only on the difference of the values of f at the endpoints of the curve, so is independent of the choice of f. By the Poincaré lemma, every closed 1-form is locally exact, so this allows ∫γ ω to be computed as a sum of differences of this kind and for the integral of closed 1-forms to be extended to continuous paths: Monodromy theorem.
A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.
Similarly to prove the first formula it suffices to show that when ψ is a smooth function compactly supported in some coordinate patch.
Otherwise it can be assumed that the coordinate patch is a disk, the boundary of which cuts the curve transversely at two points.
The intersection number of two closed curves γ1, γ2 in a Riemann surface X can be defined analytically by the formula[13][14] where α1 and α2 are smooth 1-forms of compact support corresponding to γ1 and γ2.
The definition of harmonic functions and 1-forms is intrinsic and only relies on the underlying Riemann surface structure.
If, however, a conformal metric is chosen on the Riemann surface, the adjoint d* of d can be defined and the Hodge star operation extended to functions and 2-forms.
The metric structure, however, is not required for the application to the uniformization of simply connected or planar Riemann surfaces.
The theory of Sobolev spaces on T2 can be found in Bers, John & Schechter (1979), an account which is followed in several later textbooks such as Warner (1983) and Griffiths & Harris (1994).
It provides an analytic framework for studying function theory on the torus C/Z+i Z = R2 / Z2 using Fourier series, which are just eigenfunction expansions for the Laplacian –∂2/∂x2 –∂2/∂y2.
The theory developed here essentially covers tori C / Λ where Λ is a lattice in C. Although there is a corresponding theory of Sobolev spaces on any compact Riemann surface, it is elementary in this case, because it reduces to harmonic analysis on the compact Abelian group T2.
For α = (p,q) set Dα =(Dx)p (Dy)q, a differential operator of total degree |α| = p + q.
[24] The following result—reinterpreted in the next section in terms of harmonic functions and the Dirichlet principle—is the key tool for proving the uniformization theorem for simply connected, or more generally planar, Riemann surfaces.
There is an analogous result for poles of order greater than 2 where the singular part of ω has the form z–kdz with k > 2, although this condition is not invariant under holomorphic coordinate change.
[27] If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function u on X \ {P} such that u(z) – Re z−1 is harmonic near z = 0 (the point P) such that du is square integrable on the complement of a neighbourhood of P. Moreover, if h is any real-valued smooth function on X with dh square integrable and h vanishing near P, then (du,dh)=0.
But the left hand side equals –α + i∗α = −β, with β defined exactly as in the preceding section, so this coincides with the previous construction.
Weyl (1913) proved the existence of the harmonic function u by giving a direct proof of Dirichlet's principle.
In Weyl's method of orthogonal projection, Lebesgue's theory of integration had been used to realise Hilbert spaces of 1-forms in terms of measurable 1-forms, although the 1-forms to be constructed were smooth or even analytic away from their singularity.
In the preface to Weyl (1955), referring to the extension of his method of orthogonal projection to higher dimensions by Kodaira (1949), Weyl writes: In Kodaira (2007), after giving a brief exposition of the method of orthogonal projection and making reference to Weyl's writings,[31] Kodaira explains: The methods of Hilbert spaces, Lp spaces and measure theory appear in the non-classical theory of Riemann surfaces (the study of moduli spaces of Riemann surfaces) through the Beltrami equation and Teichmüller theory.
Indeed, once this is proved, a sum of 1-forms for a chain of sufficiently close points between A and B will provide the required 1-form, since the intermediate singular terms will cancel.
The methods of group representation theory imply the operator ∆ is G-invariant, so that its fundamental solution is given by right convolution by a function on K \ G / K.[33][34] Thus in these cases Poisson's equation can be solved by an explicit integral formula.
Donaldson (2011) proves this directly for simply connected surfaces and uses it to deduce the uniformization theorem.