Differential forms on a Riemann surface

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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 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. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1940). 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 z = x + iy is holomorphic as a function of w = u + iv, then by the Cauchy–Riemann equations xu = yv and yu = – xv. In the new coordinates

so that

proving the claimed invariance.[1]

Note that for 1-forms ω1 = p1 dx + q1 dy and ω2 = p2 dx + q2 dy

In particular if ω = p dx + q dy 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 (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.[2] Thus if ω is a smooth 1-form with dω = 0 then in some open neighbourhood of a given point there is a smooth function f such that ω = df 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 Ω = dω in that neighbourhood.

If ω = p dx + q dy is a closed 1-form on (a,b) × (c,d), then py = qx. If ω = df then p = fx and q = fy. Set

so that gx = p. Then h = fg must satisfy hx = 0 and hy = qgy. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

and hence

Similarly if Ω = r dxdy then Ω = d(f dx + g dy) with gxfy = r. Thus a solution is given by f = 0 and

Comment on differential forms with compact support. Note that if ω has compact support, so vanishes outside some smaller rectangle (a1,b1) × (c1,d1) with a < a1 < b1 <b and c < c1 < d1 < d, 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 or 2-forms; but, since there is some choices for the solution, a little more care has to be taken in making those choices.[3]

In fact if Ω has compact support on (a,b) × (c,d) and if furthermore ∬ Ω = 0, then Ω = dω with ω a 1-form of compact support on (a,b) × (c,d). Indeed, Ω must have support in some smaller rectangle (a1,b1) × (c1,d1) with a < a1 < b1 <b and c < c1 < d1 < d. So r(x, y) vanishes for xa1 or xb1 and for yc1 or yd1. Let h(y) be a smooth function supported in (c1,d1) with d
c
h(t) dt = 1
. Set k(x) = ∫d
c
r(x,y) dy
: it is a smooth function supported in (a1,b1). Hence R(x,y) = r(x,y) − k(x)h(y) is smooth and supported in (a1,b1) × (c1,d1). It now satisfies d
c
R(x,y) dy ≡ 0
. Finally set

Both P and Q are smooth and supported in (a1,b1) × (c1,d1) with Py =R and Qx(x,y)= k(x)h(y). Hence ω = −P dx + Q dy is a smooth 1-form supported in (a1,b1) × (c1,d1) 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(x,y) dxdy, then |Ω| is the density |f(x,y)| dxdy, 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 atb 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 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), atb, 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 ∫γ ω can be extended to any continuous path γ(t), a ≤ t ≤ b so that it is invariant under any homotopy of paths keeping the end points fixed.[4]

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.[5] It may be assumed that [a,b] is broken up into finitely many closed intervals Ki = [ti − 1,ti] with t0 = a and tn = b so that γ(Ki) ⊂ Ui. From the above if γ is piecewise smooth
Now γ(ti) lies in the open set UiUi+1, hence in a connected open component Vi. The difference gi = fifi−1 satisfies dgi = 0, 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 ∫γ ω. The definition is independent of choices: for the curve γ can be uniformly approximated by piecewise smooth curves δ so close that δ(Ki) ⊂ Ui for all i; the formula above then equals ∫δ ω 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 ∫γ df = f(γ(b)) − f(γ(a)), 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 f(z) = ∫δ ω. The assumption implies that f is independent of the path. To check that df = ω, it suffices to check this locally. Fix z0 and take a path δ1 from w to z0. Near z0 the Poincaré lemma implies that ω =dg for some smooth function g defined in a neighbourhood of z0. If δ2 is a path from z0 to z, then f(z) = ∫δ1 ω + ∫δ2 ω = ∫δ1 ω + g(z) - g(z0), so f differs from g by a constant near z0. Hence df = dg = ω near z0.

A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.[6]

In fact the integral is already known to vanish for an exact form, so it suffices to shows that if ∫γ ω = 0 for all piecewise smooth closed Jordan curves γ then ∫γ ω = 0 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.[7] 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.[8]

Similar formulas hold on a Riemann surface X and can be deduced from the classical formulas using partitions of unity.[9] Thus if UX 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 ∫X dω = ∑ ∫X di ω) = 0, 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 = −ddf. 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 ∫γ ω = ∫X ω ∧ α for any closed smooth 1-form ω on X.[10][11]

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 Γ(t, s) of the annulus S1 x (−1,1) into X such that Γ(t,0) = γ(t). 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 s < 0 and 1 for s ≥ 0. Thus g has a jump discontinuity across γ, although its differential dg is smooth with compact support. But then, setting α = −dg, it follows from Green's formula applied to the annulus γ x [0,ε] that

Corollary 1. A closed smooth 1-form ω is exact if and only if ∫X ω ∧ α = 0 for all smooth 1-forms α of compact support.[12]

In fact if ω is exact, it has the form df for f smooth, so that ∫X ω ∧ α = ∫X df ∧ α = ∫X d(f α) = 0 by Green's theorem. Conversely, if ∫X ω ∧ α = 0 for all smooth 1-forms α of compact support, the duality between Jordan curves and 1-forms implies that the integral of ω around any closed Jordan curve is zero and hence that ω is exact.

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 ∫γ ω = ∫X ω ∧ α 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.

In fact γ is homotopic to a piecewise smooth closed curve δ, so that ∫ γ ω = ∫δ ω. On the other hand there are finitely many piecewise smooth Jordan curves δi such that δ ω = ∑ ∫δi ω. The result for δi thus implies the result for γ. If β is another form with the same property, the difference α − β satisfies ∫X ω ∧ (α − β) = 0 for all closed smooth 1-forms ω. So the difference is exact by Corollary 1. Finally, if U is any neighbourhood of the image of γ, then the last result follows by applying first assertion to γ and U in place of γ and X.

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[13][14]

where α1 and α2 are smooth 1-forms of compact support corresponding to γ1 and γ2. From the definition it follows that I1, γ2) = − I2, γ1). Since αi can be taken to have its support in a neighbourhood of the image of γi, it follows that I1 , γ2) = 0 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 dγ1dγ2 has the same or opposite sign to dxdy = −i/2 dzdz, for a local holomorphic parameter z = x + iy.[15]

Indeed, by homotopy invariance, it suffices to check this for smooth Jordan curves with nowhere vanishing derivatives. The α1 can be defined by taking α1df with f of compact support in a neighbourhood of the image of γ1 equal to 0 near the left hand side of γ1, 1 near the right hand side of γ1 and smooth off the image of γ1. Then if the points of intersection of γ2(t) with γ1 occur at t = t1, ...., tm, then
This gives the required result since the jump f∘γ2(ti+) − f∘γ2(ti−) is + 1 for a positive crossing and −1 for a negative crossing.

Holomorphic and harmonic 1-forms

A holomorphic 1-form ω is one that in local coordinates is given by an expression f(z) 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 '(z) 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 = ddh = (hxx + hyy) dxdy so that h is harmonic.[16]

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 (see below), 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 T2

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. 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.[17][18]

Let T2 = {(eix,eiy: x, y ∈ [0,2π)} = R2/Z2 = C/Λ where Λ = Z + i Z. For λ = m + i n ≅ (m,n) in Λ, set eλ (x,y) = ei(mx + ny). Furthermore, set Dx= -i∂/∂x and Dy = -i∂/∂y. For α = (p,q) set Dα =(Dx)p (Dy)q, a differential operator of total degree |α| = p + q. Thus Dαeλ = λα eλ, where λα =mpnq. The (eλ) form an orthonormal basis in C(T2) for the inner product (f,g) = (2π)−2f(x,y) g(x,y) dx dy, so that (∑ aλ eλ, ∑ bμ eμ) = ∑ aλbλ.

For f in C(T'2) and k an integer, define the kth Sobolev norm by

The associated inner product

makes C(T2) into an inner product space. Let Hk(T2) 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 (∑ aλ eλ—with respect to the kth Sobolev norm, so that Hk(T2) = {∑ aλ eλ : ∑ |aλ|2(1 + |λ|2)k < ∞} with inner product

(∑ aλ eλ, ∑ bμ eμ)(k) = ∑ aλbλ (1 + |λ|2)k.

As explained below, the elements in the intersection H(T2) = Hk(T2) are exactly the smooth functions on T2; elements in the union H−∞(T2) = Hk(T2) are just distributions on T2 (sometimes referred to as "periodic distributions" on R2).[19]

The following is a (non-exhaustive) list of properties of the Sobolev spaces.

  • Differentiability and Sobolev spaces. Ck(T2) ⊂ Hk(T2) for k ≥ 0 since, using the binomial theorem to expand (1 + |λ|2)k,
  • Differential operators. Dα Hk(T2) ⊂ Hk−|α|(T2) and Dα defines a bounded linear map from Hk(T2) to Hk−|α|(T2). The operator I + Δ defines a unitary map of Hk+2(T2) onto Hk(T2); in particular (I + Δ)k defines a unitary map of Hk(T2) onto Hk(T2) for k ≥ 0.
The first assertions follow because Dα eλ = λα eλ and |λα| ≤ |λ||α| ≤ (1 + |λ|2)|α|/2. The second assertions follow because I + Δ acts as multiplication by 1 + |λ|2 on eλ.
  • Duality. For k ≥ 0, the pairing sending f, g to (f,g) establishes a duality between Hk(T2) and Hk(T2).
This is a restatement of the fact that (I + Δ)k establishes a unitary map between these two spaces, because (f,g) = ((I + Δ)kf,g)(−k).
  • Multiplication operators. If h is a smooth function then multiplication by h defines a continuous operator on Hk(T2).
For k ≥ 0, this follows from the formula for ||f||2
(k)
above and the Leibniz rule. Continuity for Hk(T2) follows by duality, since (f,hg) = (hf,g).
  • Sobolev spaces and differentiability (Sobolev's embedding theroem). For k ≥ 0, Hk+2(T2) ⊂ Ck(T2) and sup|α|≤k |Dαf| ≤ Ck ⋅ ||f||(k+2).
The inequalities for trigonometric polynomials imply the containments. The inequality for k = 0 follows from
by the Cauchy-Schwartz inequality. The first term is finite by the integral test, since ∬C (1 + |z|2)−2 dx dy = 2π ∫
0
(1 + r2)−2 r dr
< ∞ using polar coordinates. In general if |α| ≤ k, then |sup Dαf| ≤ C0 ||Dαf||2C0Cα ⋅ ||f||k+2 by the continuity properties of Dα.
  • Smooth functions. C(T2) = Hk(T2) consists of Fourier series ∑ aλ eλ such that for all k > 0, (1 + |λ|2)k |aλ| tends to 0 as |λ| tends to ∞, i.e. the Fourier coefficients aλ are of "rapid decay".
This is an immediate consequence of the Sobolev embedding theorem.
  • Inclusion maps (Rellich's compactness theorem). If k > j, the space Hk(T2) is a subspace of Hj(T2) and the inclusion Hk(T2) Hj(T2) is compact.
With respect to the natural orthonormal bases, the inclusion map becomes multiplication by (1 + |λ|2)−(kj)/2. It is therefore compact because it is given by a diagonal matrix with diagonal entries tending to zero.
  • Elliptic regularity (Weyl's lemma). Suppose that f and u in H−∞(T2) = Hk(T2) satisfy ∆u = f. Suppose also that ψ f is a smooth function for every smooth function ψ vanishing off a fixed open set U in T2; then the same is true for u. (Thus if f is smooth off U, so is u.)
By the Leibniz rule Δ(ψu) = (Δψ) u + 2(ψxux + ψyuy) + ψ Δu, so ψu = (I + Δ)−1u + (Δψ) u + 2(ψxux + ψyuy) + ψf]. If it is known that φu lies in Hk(T2) for some k and all φ vanishing off U, then differentiating shows that φux and φuy lie in Hk−1(T2). The square-bracketed expression therefore also lies in Hk−1(T2). The operator (I + Δ)−1 carries this space onto Hk+1(T2), so that ψu must lie in Hk+1(T2). Continuing in this way, it follows that ψu lies in Hk(T2) = C(T2).
  • Hodge decomposition on functions. H0(T2) = ∆ H2(T2) ker ∆ and C(T2) = ∆ C(T2) ker ∆.
Identifying H2(T2) with L2(T2) = H0(T2) using the unitary operator I + Δ, the first statement reduces to proving that the operator T = ∆(I + Δ)−1 satisfies L2(T2) = im T ker T. This operator is bounded, self-adjoint and diagonalized by the orthonormal basis eλ with eigenvalue |λ|2(1 + |λ|2)−1. The operator T has kernel C e0 (the constant functions) and on (ker T) = im T it has a bounded inverse given by S eλ = |λ|−2(1 + |λ|2) eλ for λ ≠ 0. So im T must be closed and hence L2(T2) = (ker T) ker T = im T ker T. Finally if f = ∆g + h with f in C(T2), g in H2(T2) and h constant, g must be smooth by Weyl's lemma.[20]
  • Hodge theory on T2. Let Ωk(T2) be the space of smooth k-forms for 0 ≤ k ≤ 2. Thus Ω0(T2) = C(T2), Ω1(T2) = C(T2) dx C(T2) dy and Ω2(T2) = C(T2) dxdy. The Hodge star operation is defined on 1-forms by ∗(p dx + q dy) = −q dx + p dy. This definition is extended to 0-forms and 2-forms by *f = f dxdy and *(g dxdy) = g. Thus ** = (−1)k on k-forms. There is a natural complex inner product on Ωk(T2) defined by
Define δ = −∗d. Thus δ takes Ωk(T2) to Ωk−1(T2), annihilating functions; it is the adjoint of d for the above inner products, so that δ = d*. Indeed by the Green-Stokes formula[21]
The operators d and δ = d* satisfy d2 = 0 and δ2 = 0. The Hodge Laplacian on k-forms is defined by k = (d + d*)2 = dd* + d*d. From the definition 0 f = ∆f. Moreover 1(p dx+ q dy) =(∆p)dx + (∆q)dy and 2(f dxdy) = (∆f)dxdy. This allows the Hodge decomposition to be generalised to include 1-forms and 2-forms:
  • Hodge theorem. Ωk(T2) = ker d ker d im d im ∗d = ker d ker d* im d im d*. In the Hilbert space completion of Ωk(T2) the orthogonal complement of im d im ∗d is ker d ker d, the finite-dimensional space of harmonic k-forms, i.e. the constant k-forms. In particular in Ωk(T2) , ker d / im d = ker d ker d*, the space of harmonic k-forms. Thus the de Rham cohomology of T2 is given by harmonic (i.e. constant) k-forms.
From the Hodge decomposition on functions, Ωk(T2) = ker ∆k im ∆k. Since ∆k = dd* + d*d, ker ∆k = ker d ker d*. Moreover im (dd* + d*d) ⊊ im d im d*. Since ker d ker d* is orthogonal to this direct sum, it follows that Ωk(T2) = ker d ker d* im d im d*. The last assertion follows because ker d contains ker d ker d* im d and is orthogonal to im d* = im ∗d.

Hilbert space of 1-forms

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 2-dimensional space of constant 1-forms). The method of orthogonal projection of Weyl (1940) 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 Ω1
c
(X) denote the space of continuous 1-forms with compact support. It admits the complex inner product

for α and β in Ω1
c
(X). Let H denote the Hilbert space completion of Ω1
c
(X). 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
c
(X) and H2 denote the closure of ∗d C
c
(X). Since (df,∗dg) = ∫X dfdg = ∫X d (f dg) = 0, these are orthogonal subspaces. Let H0 denote the orthogonal complement (H1 H2) = H
1
H
2
.[22]

Theorem (Hodge−Weyl decomposition). 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 ∧ ∗ω < ∞.

  • Every square integrable continuous 1-form lies in H.
The space of continuous 1-forms of compact support is contained in the space of square integrable continuous 1-forms. They are both inner product spaces for the above inner product. So it suffices to show that any square integrable continuous 1-form can be approximated by continuous 1-forms of compact support. Let ω be a continuous square integrable 1-form, Thus the positive density Ω = ω ∧ ∗ω is integrable and there are continuous functions of compact support ψn with 0 ≤ ψn ≤ 1 such that ∫X ψn Ω tends to ∫X Ω = ||ω||2. Let φn = 1 − (1 − ψn)1/2, a continuous function of compact support with 0 ≤ φn ≤ 1. Then ωn = φn ⋅ ω tends to ω in H, since ||ω − ωn||2 = ∫X (1 − ψn) Ω tends to 0.
  • If ω in H is such that ψ ⋅ ω is continuous for every ψ in Cc(X), then ω is a square integrable continuous 1-form.
Note that the multiplication operator m(φ) given by m(φ)α = φ ⋅ α for φ in Cc(X) and α in Ω1
c
(X) satisfies ||m(φ)α|| ≤ ||φ|| ||α||, where ||φ|| = sup |φ|. Thus m(φ) defines a bounded linear operator with operator norm ||m(φ)|| ≤ ||φ||. It extends continuously to a bounded linear operator on H with the same operator norm. For every open set U with compact closure, there is a continuous function φ of compact support with 0 ≤ φ ≤ 1 with φ ≅ 1 on U. Then φ ⋅ ω is continuous on U so defines a unique continuous form ωU on U. If V is another open set intersecting U, then ωU = ωV on U V: in fact if z lies in U V and ψ in Cc(U V) ⊂ Cc(X) with ψ = 1 near z, then ψ ⋅ ωU = ψ ⋅ ω = ψ ⋅ ωV, so that ωU = ωV near z. Thus the ωU's patch together to give a continuous 1-form ω0 on X. By construction, ψ ⋅ ω = ψ ⋅ ω0 for every ψ in Cc(X). In particular for φ in Cc(X) with 0 ≤ φ ≤ 1, ∫ φ ⋅ ω0 ∧ ∗ω0 = ||φ1/2 ⋅ ω0||2 = ||φ1/2 ⋅ ω||2 ≤ ||ω||2. So ω0 ∧ ∗ω0 is integrable and hence ω0 is square integrable, so an element of H. On the other hand ω can be approximated by ωn in Ω1
c
(X). Take ψn in Cc(X) with 0 ≤ ψn ≤ 1 with ψn ⋅ ωn = ωn. Since real-valued continuous functions are closed under lattice operations. it can further be assumed that ∫ ψ2
n
ω0 ∧ ∗ω0, and hence ∫ ψn ω0 ∧ ∗ω0, increase to ||ω0||2. But then ||ψn ⋅ ω − ω|| and ||ψn ⋅ ω0 − ω0|| tend to 0. Since ψn ⋅ ω = ψn ⋅ ω0, this shows that ω = ω0.
  • Every square integrable harmonic 1-form ω lies in H0.
This is immediate because ω lies in H and, for f a smooth function of compact support, (df,ω) = ∫X df ∧ ∗ω = −∫X f d∗ω = 0 and (∗df,ω) = ∫X df ∧ ω = − ∫X f dω = 0.
  • Every element of H0 is given by a square integrable harmonic 1-form.
Let ω be an element of H0 and for fixed p in X fix a chart U in X containing p which is conformally equivalent by a map f to a disc DT2 with f(0) = p. The identification map from Ω1
c
(U) onto Ω1
c
(D) and hence into Ω1(T2) preserves norms (up to a constant factor). Let K be the closure of Ω1
c
(U) in H. Then the above map extends uniquely to an isometry T of K into H0(T2)dx H0(T2)dy. Moreover if ψ is in C
c
(U) then T m(ψ) = m(ψ ∘ f) T. The identification map T is also compatible with d and the Hodge star operator. Let D1 be a smaller concentric disk in T2 and set V = f(V). Take φ in C
c
(U) with φ ≡ 1 on V. Then (m(φ) ω,dh) = 0 = (m( φ) ω,∗dh) for h in C
c
(V). Hence, if ω1 = m(φ)ω and ω2 = T1), then (ω2, dg) = 0 = (ω2, ∗dg) for g in C
c
(D1)
.
Write ω2 = a dx + b dy with a and b in H0(T2). The conditions above imply (dω1, ∗g) = 0 = (d∗ ω1, ∗g). Replacing ∗g by dω3 with ω3 a smooth 1-form supported in D1, it follows that ∆1 ω2 = 0 on D1. Thus ∆a = 0 = ∆b on D1. Hence by Weyl's lemma, a and b are harmonic on D1. In particular both of them, and hence ω2, are smooth on D1; and dω2 = 0 = d∗ω2 on D1. Transporting these equations back to X, it follows that ω1 is smooth on V and dω1 = 0 = d∗ω1 on V. Since ω1 = m(φ)ω and p was an arbitrary point, this implies in particular that m(ψ)ω is continuous for every ψ in Cc(X). So ω is continuous and square integrable.
But then ω is smooth on V and dω = 0 = d∗ω on V. Again since p was arbitrary, this implies ω is smooth on X and dω = 0 = d∗ω on X, so that ω is a harmonic 1-form on X.

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
0
= H3H4 and that these subspaces are interchanged by complex conjugation. The smooth 1-forms in H1, H2, H3 or H4 have a simple description.[23]

  • A smooth 1-form in H1 has the form df for f smooth.
  • A smooth 1-form in H2 has the form ∗df for f smooth.
  • A smooth 1-form in H3 has the form f for f smooth.
  • A smooth 1-form in H3 has the form f for f smooth.
In fact, in view of the decompositions of H
0
and its invariance under the Hodge star operation, it suffices to prove the first of these assertions. Since H1 is invariant under complex conjugation, it may be assumed that α is a smooth real 1-form in H1. It is therefore a limit in H1 of forms dfn with fn smooth of compact support. The 1-form α must be closed since, for any real-valued f in C
c
(X),
so that dα = 0. To prove that α is exact it suffices to prove that ∫X α ∧ ∗β = 0 for any smooth closed real 1-form β of compact support. But by Green's formula

The above characterisations have an immediate corollary:

  • A smooth 1-form α in H
    0
    can be decomposed uniquely as α = da + ∗db = f + g, with a, b, f and g smooth and all the summands square integrable.

Combined with the previous Hodge–Weyl decomposition and the fact that an element of H0 is automatically smooth, this immediately implies:

Theorem (smooth Hodge–Weyl decomposition). If α is a smooth square integrable 1-form then α can be written uniquely as α = ω + da + *db = ω + f + g with ω harmonic, square integrable and a, b, f, g smooth with square integrable differentials.[24]

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 \ {P}.[25]

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 zkdz with k > 2, although this condition is not invariant under holomorphic coordinate change.

To prove uniqueness, note that if ω1 and ω2 are two solutions then their difference ω = ω1 − ω2 is a square integrable holomorphic 1-form which is exact on X \ {P}. Thus near P, ω = f(z) dz with f holomorphic near z = 0. There is a holomorphic function g on X \ {P} such that ω = dg there. But then g must coincide with a primitive of f near z = 0, so that ω = dg everywhere. But then ω lies in H0H1 = (0), i.e. ω = 0.
To prove existence, take a bump function 0 ≤ ψ ≤ 1 in C
c
(X) with support in a neighbourhood of P of the form |z| < ε and such that ψ ≡ 1 near P . Set
so that α equals z–2dz near P, vanishes off a neighbourhood of P and is exact on X \ {P}. Let β = α − i∗α, a smooth (0,1) form on X, vanishing near z =0, since it is a (1,0) form there, and vanishing off a larger neighbourhood of P. By the smooth Hodge−Weyl decomposition, β can be decomposed as β = ω0 + daida with ω0 a harmonic and square integrable (0,1) form and a smooth with square integrable differential. Now set γ = α – da = ω0 + i∗α − ida and ω = Re γ + i∗ Re γ. Then α is exact on X \ {P}; hence so is γ, as well as its real part, which is also the real part of ω. Near P, the 1-form ω differs from z–2dz by a smooth (1,0) form. It remains to prove that ω = 0 on X \ {P}; or equivalently that Re γ is harmonic on X \ {P}. In fact γ is harmonic on X \ {P}; for dγ = dα − d(da) = 0 on X \ {P} because α is exact there; and similarly d∗γ = 0 using the formula γ = ω0 + i∗α − ida and the fact that ω0 is harmonic.

Corollary of proof. [26] 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 \ {P} such that δ – Re z−2dz is harmonic near z = 0 (the point P) 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 (δ,dh) = 0.

Existence follows by taking δ = Re γ = Re ω above. Since ω = δ + i∗δ, the uniqueness of ω implies the uniqueness of δ. Alternatively if δ1 and δ2 are two solutions, their difference η = δ1 – δ2 has no singularity at P and is harmonic on X \ {P}. It is therefore harmonic in a neighbourhood of P and therefore everywhere. So η lies in H0. But also η is exact on X \ P and hence on the whole of X, so it also lies in H1. But then it must lie in H0H1 = (0), so that η = 0. Finally, if N is the closure of a neighbourhood of P disjoint from the support of h and Y = X \ N, then δ|Y lies in H0(Y) and dh lies in the space H1(Y) so that

Dirichlet's principle on a Riemann surface

Theorem.[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.

In fact this result is immediate from the theorem and corollary in the previous section. The harmonic form δ constructed there is the real part of a holomorphic form ω = dg where g is holomorphic function on X with a simple pole at P with residue -1, i.e. g(z) = –z−1 + a0 + a1z + a2 z2 + ⋅ ⋅ ⋅ near z = 0. So u = - Re g gives a solution with the claimed properties since δ = −du and hence (du,dh) = −(δ,dh) = 0.

This result can be interpreted in terms of Dirichlet's principle.[28][29][30] Let DR be a parametric disk |z| < R about P (the point z = 0) with R > 1. Let α = −dz−1), where 0 ≤ ψ ≤ 1 is a bump function supported in D = D1, identically 1 near z = 0. Let α1 = −χD(z) Re d(z−1) where χD is the characteristic function[disambiguation needed] 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

F(ξ) = ||γ1 − γ – ξ||

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 (η12) 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 ∗(α1 – α) onto H2. On the other hand, ∗α1 = −i α1, since α is a (1,0) form. Hence

1 – α) − i∗(α1 – α) = ω0 + ω1 + ω2,

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 Hurwitz & Courant (1929), Ahlfors (1947), Courant (1950), Schiffer & Spencer (1954), Pfluger (1957) and Ahlfors & Sario (1960).

Historical note. Weyl (1913) proved the existence of the harmonic function u by giving a direct proof of Dirichlet's principle. In Weyl (1940), he presented his method of orthogonal projection which has been adopted in the presentation above, following Springer (1957), but with the theory of Sobolev spaces on T2 used to prove elliptic regularity without using measure theory. In the expository texts Weyl (1955) and Kodaira (2007), 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 Weyl (1955), referring to the extension of his method of orthogonal projection to higher dimensions by Kodaira (1949), Weyl writes:

"Influenced by Kodaira's work, I have hesitated a moment as to whether I should not replace the Dirichlet principle by the essentially equivalent "method of orthogonal projection" which is treated in a paper of mine. But for reasons the explication of which would lead too far afield here, I have stuck to the old approach."

In Kodaira (2007), after giving a brief exposition of the method of orthogonal projection and making reference to Weyl's writings,[31] Kodaira explains:

"I first planned to prove Dirichlet's Principle using the method of orthogonal projection in this book. However, I did not like to have to use the concept of Lebesgue measurability only for the proof of Dirichlet's Principle and therefore I rewrote it in such a way that I did not have to."

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.

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.[32]

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 (Poisson equation). 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 fact, Ω can be written as Ω = dα with α a smooth 1-form of compact support: indeed, using partitions of unity, this reduces to the case of a smooth 2-form of compact support on a rectangle. Indeed Ω can be written as a finite sum of 2-forms each supported in a parametric rectangle and having integral zero. For each of these 2-forms the result follows from Poincaré's lemma with compact support. Writing α = ω + da + *db, it follows that Ω = d*db = ∆b.

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(2,R) and SU(2). 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. 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 ∞. Donaldson (2011) proves this directly for simply connected surfaces and uses it to deduce the uniformization theorem.[35]

See also

Notes

  1. ^ Springer 1957, p. 165
  2. ^ Napier & Ramachandran 2011, pp. 443–444
  3. ^ Donaldson 2011, pp. 70–71
  4. ^ See:
  5. ^ 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 ds2 = f(z) |dz|2, then any non-empty intersection of finitely many Ui would be geodesically convex and hence connected; see do Carmo 1976, pp. 303–305.
  6. ^ Kodaira 2007, pp. 290–292
  7. ^ Kodaira 2007, pp. 290–292
  8. ^ Kodaira 2007, pp. 251–256
  9. ^ See:
  10. ^ Kodaira 2007, pp. 292–293
  11. ^ Springer 1957, pp. 200–201
  12. ^ Kodaira 2007, p. 294
  13. ^ See:
  14. ^ Note that more generally intersection theory has also been developed separately within differential topology using Sard's theorem. See for example:
  15. ^ This makes sense if the tangent vectors to the two curves at the crossing point exist, are non-vanishing and are transverse there, i.e. are not proportional.
  16. ^ Springer 1957, pp. 168–172
  17. ^ For treatments in texts on Riemann surfaces, see:
  18. ^ For treatments in texts on partial differential equations, see for example:
  19. ^ See:
  20. ^ Note that it is easy to see directly that ∆ is an isomorphism on smooth functions orthogonal to the constants, since these are just Fourier series of rapid decay without constant term.
  21. ^ Warner 1983, pp. 220–221
  22. ^ Springer 1957, pp. 178–206
  23. ^ Springer 1957, pp. 200–201
  24. ^ Springer 1957, pp. 195–205
  25. ^ Springer 1957, pp. 209–211
  26. ^ Springer 1957, pp. 209–212
  27. ^ Springer 1957, pp. 209–212, 219
  28. ^ Springer 1957, pp. 211–212
  29. ^ Kodaira 2007, pp. 294–318
  30. ^ Weyl 1955, pp. 93–118
  31. ^ Kodaira & 312−314
  32. ^ Springer 1957, pp. 212–213
  33. ^ Helgason 2001, p. 444–449
  34. ^ Folland 1995, pp. 104–108
  35. ^ Donaldson 2011, pp. 131–143

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