Differential forms on a Riemann surface


In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms without specifying a Riemannian metric. 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 in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later 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. Later they supplied the analytic foundations for the harmonic integrals of. This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.

Hodge star on 1-forms

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.
Indeed, if is holomorphic as a function of,
then by the Cauchy–Riemann equations and. In the new coordinates
so that
proving the claimed invariance.
Note that for 1-forms and
In particular if then
Note that in standard coordinates
Recalling that
so that
The decomposition is independent of the choice of local coordinate. The 1-forms with only a component are called forms; those with only a component are called forms. The operators and are called the Dolbeault operators.
It follows that
The Dolbeault operators can similarly be defined on 1-forms and as zero on 2-forms. They have the properties

Poincaré lemma

On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. Thus if ω is a smooth 1-form with then in some open neighbourhood of a given point there is a smooth function f such that in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form ω defined in some open neighbourhood of a given point such that in that neighbourhood.
If is a closed 1-form on, then. If then and. Set
so that. Then must satisfy and. The right hand side here is independent of x since its partial derivative with respect to x is 0. So
and hence
Similarly, if then with. Thus a solution is given by and
Comment on differential forms with compact support. Note that if ω has compact support, so vanishes outside some smaller rectangle with and, then the same is true for the solution f. So the Poincaré lemma for 1-forms holds with this additional conditions of compact support.
A similar statement is true or 2-forms; but, since there is some choices for the solution, a little more care has to be taken in making those choices.
In fact if Ω has compact support on and if furthermore, then with ω a 1-form of compact support on. Indeed, Ω must have support in some smaller rectangle with and. So vanishes for or and for or. Let h be a smooth function supported in with. Set : it is a smooth function supported in . Hence is smooth and supported in. It now satisfies. Finally set
Both P and Q are smooth and supported in with and. Hence is a smooth 1-form supported in with

Integration of 2-forms

If Ω is a continuous 2-form of compact support on a Riemann surface X, its support K can be covered by finitely many coordinate charts Ui and there is a partition of unity χi of smooth non-negative functions with compact support such that ∑ χi = 1 on a neighbourhood of K. Then the integral of Ω is defined by
where the integral over Ui has its usual definition in local coordinates. The integral is independent of the choices here.
If Ω has the local representation f dx ∧ dy, then |Ω| is the density |f| dx ∧ dy, which is well defined and satisfies |∫X Ω| ≤ ∫X |Ω|. If Ω is a non-negative continuous density, not necessarily of compact support, its integral is defined by
If Ω is any continuous 2-form it is integrable if ∫X |Ω| < ∞. In this case, if ∫X |Ω| = lim ∫X ψn |Ω|, then ∫X Ω can be defined as lim ∫X ψn Ω. The integrable continuous 2-forms form a complex normed space with norm ||Ω||1 = ∫X |Ω|.

Integration of 1-forms along paths

If ω is a 1-form on a Riemann surface X and γ for is a smooth path in X, then the mapping γ induces a 1-form γ∗ω on . The integral of ω along γ is defined by
This definition extends to piecewise smooth paths γ by dividing the path up into the finitely many segments on which it is smooth. In local coordinates if and then
so that
Note that if the 1-form ω is exact on some connected open set U, so that for some smooth function f on U, and γ,, 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. If ω is a closed 1-form, the integral can be extended to any continuous path γ, so that it is invariant under any homotopy of paths keeping the end points fixed.
The same argument shows that a homotopy between closed continuous loops does not change their integrals over closed 1-forms. Since, the integral of an exact form over a closed loop vanishes.
Conversely if the integral of a closed 1-form ω over any closed loop vanishes, then the 1-form must be exact.
A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.
The above argument also shows that given a continuous Jordan curve γ, there is a finite set of simple smooth Jordan curves γi with nowhere zero derivatives such that
for any closed 1-form ω. Thus to check exactness of a closed form it suffices to show that the vanishing of the integral around any regular closed curve, i.e. a simple smooth Jordan curve with nowhere vanishing derivative.
The same methods show that any continuous loop on a Riemann surface is homotopic to a smooth loop with nowhere zero derivative.

Green–Stokes formula

If U is a bounded region in the complex plane with boundary consisting of piecewise smooth curves and ω is a 1-form defined on a neighbourhood of the closure of U, then the Green–Stokes formula states that
In particular if ω is a 1-form of compact support on C then
since the formula may be applied to a large disk containing the support of ω.
Similar formulas hold on a Riemann surface X and can be deduced from the classical formulas using partitions of unity. Thus if is a connected region with compact closure and piecewise smooth boundary ∂U and ω is a 1-form defined on a neighbourhood of the closure of U, then the Green–Stokes formula states that
Moreover, if ω is a 1-form of compact support on X then
To prove the second formula take a partition of unity ψi supported in coordinate charts covering the support of ω. Then, by the planar result. Similarly to prove the first formula it suffices to show that
when ψ is a smooth function compactly supported in some coordinate patch. If the coordinate patch avoids the boundary curves, both sides vanish by the second formula above. Otherwise it can be assumed that the coordinate patch is a disk, the boundary of which cuts the curve transversely at two points. The same will be true for a slightly smaller disk containing the support of ψ. Completing the curve to a Jordan curve by adding part of the boundary of the smaller disk, the formula reduces to the planar Green-Stokes formula.
The Green–Stokes formula implies an adjoint relation for the Laplacian on functions defined as Δf = −d∗df. This gives a 2-form, given in local coordinates by the formula
Then if f and g are smooth and the closure of U is compact
Moreover, if f or g has compact support then

Duality between 1-forms and closed curves

Theorem. If γ is a continuous Jordan curve on a Riemann surface X, there is a smooth closed 1-form α of compact support such that for any closed smooth 1-form ω on X.
Corollary 1. A closed smooth 1-form ω is exact if and only if for all smooth 1-forms α of compact support.
Corollary 2. If γ is a continuous closed curve on a Riemann surface X, there is a smooth closed 1-form α of compact support such that for any closed smooth 1-form ω on X. The form α is unique up to adding an exact form and can be taken to have support in any open neighbourhood of the image of γ.

Intersection number of closed curves

The intersection number of two closed curves γ1, γ2 in a Riemann surface X can be defined analytically by the formula
where α1 and α2 are smooth 1-forms of compact support corresponding to γ1 and γ2. From the definition it follows that. Since αi can be taken to have its support in a neighbourhood of the image of γi, it follows that if γ1 and γ2 are disjoint. By definition it depends only on the homotopy classes of γ1 and γ2.
More generally the intersection number is always an integer and counts the number of times with signs that the two curves intersect. A crossing at a point is a positive or negative crossing according to whether dγ1 ∧ dγ2 has the same or opposite sign to, for a local holomorphic parameter z = x + iy.

Holomorphic and harmonic 1-forms

A holomorphic 1-form ω is one that in local coordinates is given by an expression f dz with f holomorphic. Since it follows that dω = 0, so any holomorphic 1-form is closed. Moreover, since ∗dz = -i dz, ω must satisfy ∗ω = -iω. These two conditions characterize holomorphic 1-forms. For if ω is closed, locally it can be written as dg for some g, The condition ∗dg = i dg forces, so that g is holomorphic and dg = g ' dz, so that ω is holomorphic.
Let ω = f dz be a holomorphic 1-form. Write ω = ω1 + iω2 with ω1 and ω2 real. Then dω1 = 0 and dω2 = 0; and since ∗ω = -iω, ∗ω1 = ω2. Hence d∗ω1 = 0. This process can clearly be reversed, so that there is a one-one correspondence between holomorphic 1-forms and real 1-forms ω1 satisfying dω1 = 0 and d∗ω1 = 0. Under this correspondence, ω1 is the real part of ω while ω is given by ω= ω1 + i∗ω1. Such forms ω1 are called harmonic 1-forms. By definition ω1 is harmonic if and only if ∗ω1 is harmonic.
Since holomorphic 1-forms locally have the form df with f a holomorphic function and since the real part of a holomorphic function is harmonic, harmonic 1-forms locally have the form dh with h a harmonic function. Conversely if ω1 can be written in this way locally, d∗ω1 = d∗dh = dx∧dy so that h is harmonic.
Remark. 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 Hodge Laplacian can be defined on k-forms as ∆k = dd* +d*d and then a function f or a 1-form ω is harmonic if and only if it is annihilated by the Hodge Laplacian, i.e. ∆0f = 0 or ∆1ω = 0. The metric structure, however, is not required for the application to the uniformization of simply connected or planar Riemann surfaces.

Sobolev spaces on '''T'''2

The theory of Sobolev spaces on can be found in, an account which is followed in several later textbooks such as and. 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. 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. Classical approaches to Weyl's lemma use harmonic analysis on the non-compact Abelian group C = R2, i.e. the methods of Fourier analysis, in particular convolution operators and the fundamental solution of the Laplacian.
Let T2 = = R2/Z2 = C/Λ where Λ = Z + i Z.
For λ = m + i n ≅ in Λ, set. Furthermore, set Dx= -i∂/∂x and Dy = -i∂/∂y. For α = set Dα =p q, a differential operator of total degree |α| = p + q. Thus, where. The form an orthonormal basis in C for the inner product, so that.
For f in C∞ and k an integer, define the kth Sobolev norm by
The associated inner product
makes C∞ into an inner product space. Let Hk be its Hilbert space completion. It can be described equivalently as the Hilbert space completion of the space of trigonometric polynomials—that is finite sums —with respect to the kth Sobolev norm, so that Hk = with inner product
As explained below, the elements in the intersection H∞ = Hk are exactly the smooth functions on T2; elements in the union H−∞ = Hk are just distributions on T2.
The following is a list of properties of the Sobolev spaces.
In the case of the compact Riemann surface C / Λ, the theory of Sobolev spaces shows that the Hilbert space completion of smooth 1-forms can be decomposed as the sum of three pairwise orthogonal spaces, the closure of exact 1-forms df, the closure of coexact 1-forms ∗df and the harmonic 1-forms. The method of orthogonal projection of put Riemann's approach to the Dirichlet principle on sound footing by generalizing this composition to arbitrary Riemann surfaces.
If X is a Riemann surface Ω denote the space of continuous 1-forms with compact support. It admits the complex inner product
for α and β in Ω. Let H denote the Hilbert space completion of Ω. Although H can be interpreted in terms of measurable functions, like Sobolev spaces on tori it can be studied directly using only elementary functional analytic techniques involving Hilbert spaces and bounded linear operators.
Let H1 denote the closure of d C and H2 denote the closure of ∗d C. Since, these are orthogonal subspaces. Let H0 denote the orthogonal complement
⊥ = H H.
Theorem. H = H0 H1 H2. The subspace H0 consists of square integrable harmonic 1-forms on X, i.e. 1-forms ω such that dω = 0, d∗ω = 0 and ||ω||2 = ∫X ω ∧ ∗ < ∞.
From the formulas for the Dolbeault operators and, it follows that
where both sums are orthogonal. The two subspaces in the second sum correspond to the ±i eigenspaces of the Hodge ∗ operator. Denoting their closures by H3 and H4, it follows that H = H3 ⊕ H4 and that these subspaces are interchanged by complex conjugation. The smooth 1-forms in H1, H2, H3 or H4 have a simple description.
The above characterisations have an immediate corollary:
Combined with the previous Hodge–Weyl decomposition and the fact that an element of H0 is automatically smooth, this immediately implies:
Theorem. If α is a smooth square integrable 1-form then α can be written uniquely as with ω harmonic, square integrable and smooth with square integrable differentials.

Holomorphic 1-forms with a double pole

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.
Theorem. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique holomorphic differential 1-form ω with a double pole at P, so that the singular part of ω is z−2dz near P, and regular everywhere else, such that ω is square integrable on the complement of a neighbourhood of P and the real part of ω is exact on X \.
The double pole condition is invariant under holomorphic coordinate change z z + az2 + ⋅ ⋅ ⋅. 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.
Corollary of proof. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued 1-form δ which is harmonic on X \ such that δ – Re z−2dz is harmonic near z = 0 such that δ 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 = 0.

Dirichlet's principle on a Riemann surface

Theorem. 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 \ such that u – Re z−1 is harmonic near z = 0 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 =0.
This result can be interpreted in terms of Dirichlet's principle. Let DR be a parametric disk |z| < R about P with R > 1. Let α = −d, where 0 ≤ ψ ≤ 1 is a bump function supported in D = D1, identically 1 near z = 0. Let α1 = −χD Re d where χD is the characteristic function of D. Let γ= Re α and γ1 = Re α1. Since χD can be approximated by bump functions in L2, γ1 − γ lies in the real Hilbert space of 1-forms Re H; similarly α1 − α lies in H. Dirichlet's principle states that the distance function
on Re H1 is minimised by a smooth 1-form ξ0 in Re H1. In fact −du coincides with the minimising 1-form: γ + ξ0 = -du.
This version of Dirichlet's principle is easy to deduce from the previous construction of du. By definition ξ0 is the orthogonal projection of γ1 – γ onto Re H1 for the real inner product Re on H, regarded as a real inner product space. It coincides with the real part of the orthogonal projection ω1 of α1 – α onto H1 for the complex inner product on H. Since the Hodge star operator is a unitary map on H swapping H1 and H2, ω2 = ∗ω1 is the orthogonal projection of ∗ onto H2. On the other hand, ∗α1 = −i α1, since α is a form. Hence
with ωk in Hk. 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.
Further discussion of Dirichlet's principle on a Riemann surface can be found in,,,, and.
Historical note. proved the existence of the harmonic function u by giving a direct proof of Dirichlet's principle. In, he presented his method of orthogonal projection which has been adopted in the presentation above, following, but with the theory of Sobolev spaces on T2 used to prove elliptic regularity without using measure theory. In the expository texts and, both authors avoid invoking results on measure theory: they follow Weyl's original approach for constructing harmonic functions with singularities via 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, referring to the extension of his method of orthogonal projection to higher dimensions by, Weyl writes:
In, after giving a brief exposition of the method of orthogonal projection and making reference to Weyl's writings, Kodaira explains:
The methods of Hilbert spaces, Lp spaces and measure theory appear in the non-classical theory of Riemann surfaces through the Beltrami equation and Teichmüller theory.

Holomorphic 1-forms with two single poles

Theorem. Given a Riemann surface X and two distinct points A and B on X, there is a holomorphic 1-form on X with simple poles at the two points with non-zero residues having sum zero such that the 1-form is square integrable on the complement of any open neighbourhoods of the two points.
The proof is similar to the proof of the result on holomorphic 1-forms with a single double pole. The result is first proved when A and B are close and lie in a parametric disk. 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. To construct the 1-form for points corresponding to a and b in a parametric disk, the previous construction can be used starting with the 1-form
which locally has the form

Poisson equation

Theorem. If Ω is a smooth 2-form of compact support on a Riemann surface X, then Ω can be written as Ω = ∆f where f is a smooth function with df square integrable if and only if ∫X Ω = 0.
In the case of the simply connected Riemann surfaces C, D and S= C ∪ ∞, the Riemann surfaces are symmetric spaces G / K for the groups G = R2, SL and SU. 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. Thus in these cases Poisson's equation can be solved by an explicit integral formula. It is easy to verify that this explicit solution tends to 0 at ∞, so that in the case of these surfaces there is a solution f tending to 0 at ∞. proves this directly for simply connected surfaces and uses it to deduce the uniformization theorem.