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 (or differentials) 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
Recall also that
so that
The decomposition is independent of the choice of local coordinate. The 1-forms with only a component are called (1,0) forms; those with only a component are called (0,1) 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(x,y). So the Poincaré lemma for 1-forms holds with this additional conditions of compact support.
A similar statement is true for 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(y) be a smooth function supported in (c1,d1) with . Set : it is a smooth function supported in (a1,b1). 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 Σ 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(x,y) dx ∧ dy, then |Ω| is the density |f(x,y)| 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(t) for is a smooth path in X, then the mapping γ induces a 1-form γ∗ω on [a,b]. 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 (unique up to a constant), and γ(t), , 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(t), so that it is invariant under any homotopy of paths keeping the end points fixed.See:
In fact, the image of γ is compact, so can be covered by finitely many connected open sets Ui on each of which ω can be written dfi for some smooth function fi on Ui, unique up to a constant.It is not assumed that if two Ui's intersect then their intersection is connected, as is the case for disks in the plane. Note, however, that if the Ui were chosen to be small geodesic disks for a conformal Riemannian metric, locally of the form , then any non-empty intersection of finitely many Ui would be geodesically convex and hence connected; see . It may be assumed that [a,b] is broken up into finitely many closed intervals with and so that . From the above if γ is piecewise smooth,
Now γ(ti) lies in the open set , hence in a connected open component Vi. The difference satisfies , so is a constant ci independent of γ. Hence
The formula on the right hand side also makes sense if γ is just continuous on [a,b] and can be used to define . The definition is independent of choices: for the curve γ can be uniformly approximated by piecewise smooth curves δ so close that for all i; the formula above then equals and shows the integral is independent of the choice of δ. The same argument shows that the definition is also invariant under small homotopies fixing endpoints; by compactness, it is therefore invariant under any homotopy fixing endpoints.
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.
Indeed a function f(z) can be defined on X by fixing a point w, taking any path δ from w to z and setting . The assumption implies that f is independent of the path. To check that , it suffices to check this locally. Fix z0 and take a path δ1 from w to z0. Near z0 the Poincaré lemma implies that for some smooth function g defined in a neighbourhood of z0. If δ2 is a path from z0 to z, then , so f differs from g by a constant near z0. Hence near z0.
A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.
In fact the integral is already known to vanish for an exact form, so it suffices to show that if for all piecewise smooth closed Jordan curves γ then for all closed continuous curves γ. Let γ be a closed continuous curve. The image of γ can be covered by finitely many opens on which ω is exact and this data can be used to define the integral on γ. Now recursively replace γ by smooth segments between successive division points on the curve so that the resulting curve δ has only finitely many intersection points and passes through each of these only twice. This curve can be broken up as a superposition of finitely many piecewise smooth Jordan curves. The integral over each of these is zero, so their sum, the integral over δ, is also zero. By construction the integral over δ equals the integral over γ, which therefore vanishes.
The above argument also shows that given a continuous Jordan curve γ(t), there is a finite set of simple smooth Jordan curves γi(t) 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.See: 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.
It suffices to prove this when γ is a regular closed curve. By the inverse function theorem, there is a tubular neighbourhood of the image of γ, i.e. a smooth diffeomorphism of the annulus into X such that . Using a bump function on the second factor, a non-negative function g with compact support can be constructed such that g is smooth off γ, has support in a small neighbourhood of γ, and in a sufficiently small neighbourhood of γ is equal to 0 for and 1 for . Thus g has a jump discontinuity across γ, although its differential dg is smooth with compact support. But then, setting , it follows from Green's formula applied to the annulus that
Corollary 1. A closed smooth 1-form ω is exact if and only if for all smooth 1-forms α of compact support.