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 (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

\displaystyle {\star (p\,dx+q\,dy)=-q\,dx+p\,dy.}

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 x_u = y_v and y_u = – x_v. In the new coordinates

\displaystyle {p\,dx+q\,dy=(px_{u}+qy_{u})du+(px_{v}+qy_{v})dv=p_{1}du+q_{1}dv,}

so that

\displaystyle {-q_{1}\,du+p_{1}\,dv=-(px_{v}+qy_{v})du+(px_{u}+qy_{u})dv=-q(x_{u}du+x_{v}dv)+p(y_{u}du+y_{v}dv)=-q\,dx+p\,dy,}

proving the claimed invariance.^[1]

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

\displaystyle {\omega _{1}\wedge \star \omega _{2}=(p_{1}p_{2}+q_{1}q_{2})\,dx\wedge dy=\omega _{2}\wedge \star \omega _{1}.}

In particular if ω = p dx + q dy then

\displaystyle {\omega \wedge \star \omega =(p^{2}+q^{2})\,dx\wedge dy.}

Note that in standard coordinates

\star dz=-idz,\,\,\star d{\overline {z}}=id{\overline {z}}.

Recalling that

{\partial  \over \partial z}={1 \over 2}\left({\partial  \over \partial x}-i{\partial  \over \partial y}\right),\,\,\,{\partial  \over \partial {\overline {z}}}={1 \over 2}\left({\partial  \over \partial x}+i{\partial  \over \partial y}\right),

so that

df=f_{z}\,dz+f_{\overline {z}}\,d{\overline {z}}=\partial f+{\bar {\partial }}f.

The decomposition $d=\partial +{\bar {\partial }}$ is independent of the choice of local coordinate. The 1-forms with only a $dz$ component are called (1,0) forms; those with only a $d{\overline {z}}$ component are called (0,1) forms. The operators $\partial$ and $\overline {\partial }$ are called the Dolbeault operators.

It follows that

\star df=-i\partial f+i{\bar {\partial }}f.

The Dolbeault operators can similarly be defined on 1-forms and as zero on 2-forms. They have the properties

d=\partial +{\bar {\partial }}

\partial ^{2}={\bar {\partial }}^{2}=\partial {\bar {\partial }}+{\bar {\partial }}\partial =0.

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 p_y = q_x. If ω = df then p = f_x and q = f_y. Set

\displaystyle {g(x,y)=\int _{a}^{x}p(t,y)\,dt,}

so that g_x = p. Then h = f − g must satisfy h_x = 0 and h_y = q − g_y. The right hand side here is independent of x since its partial derivative with respect to x is 0. So

\displaystyle {h(x,y)=\int _{c}^{y}q(a,s)\,ds-g(a,y)=\int _{c}^{y}q(a,s)\,ds,}

and hence

\displaystyle {f(x,y)=\int _{a}^{x}p(t,y)\,dt+\int _{c}^{y}q(a,s)\,ds.}

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

\displaystyle {g(x,y)=\int _{a}^{x}r(t,y)\,dt.}

Note that if Ω has compact support on (a,b) × (c,d) and if furthermore ∬ Ω = 0, then Ω = dω with ω of compact support.^[3] 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 x ≤ a₁ or x ≥ b₁ and for y ≤ c₁ or y ≥ d₁. Let h(y) be a smooth function supported in (c₁,d₁) with $\int d c h (t) dt = 1$ . Set $k (x) = \int d c r (x, y) dy$ : 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 $\int d c R (x, y) dy \equiv 0$ . Finally set

P(x,y)=\int _{c}^{y}R(x,y)dy,\,\,\,Q(x,y)=h(y)\int _{a}^{x}k(s)\,ds.

Both P and Q are smooth aand supported in (a₁,b₁) × (c₁,d₁) with P_y =R and Q_x(x,y)= k(x)h(y). Hence ω = −P dx + Q dy is a smooth 1-form supported in (a₁,b₁) × (c₁,d₁) with

d\omega =(Q_{x}+P_{y})\,dx\wedge dy=r\,dx\wedge dy=\Omega .

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 U_i and there is a partition of unity χ_i of smooth non-negative functions with compact support such that ∑ χ_i = 1 on a neighbourhood of K. Then the integral of Ω is defined by

\displaystyle {\int _{X}\Omega =\sum \int _{X}\chi _{i}\Omega =\sum \int _{U_{i}}\chi _{i}\Omega ,}

where the integral over U_i 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

\displaystyle {\int _{X}\Omega =\sup _{0\leq \psi \leq 1,\,\psi \in C_{c}(X)}\int _{X}\psi \Omega .}

If Ω is any continuous 2-form it is integrable if ∫_X |Ω| < ∞. In this case, if ∫_X |Ω| = lim ∫_X ψ_n |Ω|, then ∫_X Ω can be defined as lim ∫_X ψ_n 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 a ≤ t ≤ b 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

\displaystyle {\int _{\gamma }\omega =\int _{a}^{b}\gamma ^{*}\omega .}

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

\displaystyle {\gamma ^{*}\omega =p(\gamma (t)){\dot {x}}(t)\,dt+q(\gamma (t)){\dot {y}}(t)\,dt,}

so that

\displaystyle {\int _{\gamma }\omega =\int _{a}^{b}p(\gamma (t)){\dot {x}}(t)\,dt+q(\gamma (t)){\dot {y}}(t)\,dt.}

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

\displaystyle {\int _{\gamma }\omega =\int _{a}^{b}d(f(\gamma (t))=f(\gamma (b))-f(\gamma (a)).}

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 U_i on each of which ω can be written df_i for some smooth function f_i on U_i, unique up to a constant.^[5] It may be assumed that [a,b] is broken up into finitely many closed intervals K_i = [t_{i − 1},t_i] with t₀ = a and t_n = b so that γ(K_i) ⊂ U_i. From the above if γ is piecewise smooth

\displaystyle {\int _{\gamma }\omega =\sum \int _{\gamma |_{K_{i}}}\omega =\sum \int _{K_{i}}d(f_{i}\circ \gamma )=\sum _{i=1}^{n}f_{i}(\gamma (t_{i}))-f_{i}(\gamma (t_{i-1}))=f_{n}(\gamma (b))-f_{1}(\gamma (b))+\sum _{i=1}^{n-1}[f_{i}(\gamma (t_{i}))-f_{i+1}(\gamma (t_{i}))].}

Now γ(t_i) lies in the open set U_i ∩ U_i+1, hence in a connected open component V_i. The difference g_i = f_i − f_i−1 satisfies dg_i = 0, so is a constant c_i independent of γ. Hence

\displaystyle {\int _{\gamma }\omega =f_{n}(\gamma (b))-f_{1}(\gamma (a))+\sum _{i=1}^{n-1}c_{i}.}

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(K_i) ⊂ U_i 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 γ_i(t) with nowhere zero derivatives such that

\int _{\gamma }\omega =\sum _{i}\int _{\gamma _{i}}\omega

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.

Green–Stokes formula

See also: Green's theorem and Stokes' theorem

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

\displaystyle {\int _{\partial U}\omega =\int _{U}d\omega .}

In particular if ω is a 1-form of compact support on C then

\displaystyle {\int _{\bf {C}}d\omega =0,}

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 U ⊂ X 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

\displaystyle {\int _{\partial U}\omega =\int _{U}d\omega .}

Moreover, if ω is a 1-form of compact support on X then

\displaystyle {\int _{X}d\omega =0.}

To prove the second formula take a partition of unity ψ_i supported in coordinate charts covering the support of ω. Then ∫_X dω = ∑ ∫_X d(ψ_i ω) = 0, by the planar result. Similarly to prove the first formula it suffices to show that

\displaystyle {\int _{\partial U}\psi \omega =\int _{U}d(\psi \omega )}

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

\displaystyle {\Delta f=\left(-{\partial ^{2}f \over \partial x^{2}}-{\partial ^{2}f \over \partial y^{2}}\right)\,dx\wedge dy.}

Then if f and g are smooth and the closure of U is compact

\displaystyle {\int _{U}f\Delta g-g\Delta f=\int _{\partial U}g\star df-f\star dg.}

Moreover, if f or g has compact support then

\displaystyle {\int _{X}f\Delta g=\int _{X}g\Delta f.}

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 neigbourhood of the image of γ, i.e. a smooth diffeomorphism

Γ(t, s)

of the annulus

S 1 x (-1,1)

into X such that

{{{1}}}

. 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

\int _{X}\omega \wedge \alpha =\int _{X}dg\wedge \omega =\int _{\gamma \times (0,\varepsilon )}dg\wedge \omega =\int _{\gamma \times (0,\varepsilon )}d(g\omega )=\int _{\gamma }\omega .

Corollary. 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.

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 $dg=\partial _{z}g\,dz+\partial _{\overline {z}}g\,d{\overline {z}},$ 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 $\partial _{\overline {z}}g=0$ , 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 = d∗dh = (h_xx + h_yy) dx∧dy so that h is harmonic.^[13]

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 $T 2$ 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 $-\partial 2 /\partial x 2 -\partial 2 /\partial y 2$ . 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 $T 2$ . 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.^[14]^[15]

Let T² = {(e^ix,e^iy: x, y ∈ [0,2π)} = R²/Z² = C/Λ where Λ = Z + i Z. For λ = m + i n ≅ (m,n) in Λ, set $e λ (x, y) = e i (mx + ny)$ . Furthermore, set D_x= -i∂/∂x and D_y = -i∂/∂y. For α = (p,q) set D^α =(D_x)^p (D_y)^q, a differential operator of total degree |α| = p + q. Thus $D α e λ = λ α e λ$ , where $λ α = m p n q$ . The (e_λ) form an orthonormal basis in C(T²) for the inner product $(f, g) = (2π) -2 \iint f (x, y) g (x, y) dx dy$ , so that $(\sum a λ e λ, \sum b μ e μ) = \sum a λ b λ$ .

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

\displaystyle {\|f\|_{(k)}=\left(\sum |{\widehat {f}}(\lambda )|^{2}(1+|\lambda |^{2})^{k}\right)^{1/2}.}

The associated inner product

\displaystyle {(f,g)_{(k)}=\sum {\widehat {f}}(\lambda ){\overline {{\widehat {g}}(\lambda )}}(1+|\lambda |^{2})^{k}}

makes C^∞(T²) into an inner product space. Let H_k(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 $(\sum a λ e λ$ —with respect to the kth Sobolev norm, so that H_k(T²) = {∑ a_λ e_λ : ∑ |a_λ|²(1 + |λ|²)^k < ∞} with inner product

(∑ a_λ e_λ, ∑ b_μ e_μ)_(k) = ∑ a_λb_λ (1 + |λ|²)^k.

As explained below, the elements in the intersection H_∞(T²) = $\cap$ H_k(T²) are exactly the smooth functions on T²; elements in the union H_−∞(T²) = $\cup$ H_k(T²) are just distributions on T² (sometimes referred to as "periodic distributions" on R²).^[16]

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

Differentiability and Sobolev spaces. $C k (T 2) \subset H k (T 2)$ for k ≥ 0 since, using the binomial theorem to expand (1 + |λ|²)^k,

\displaystyle {\|f\|_{(k)}^{2}=\sum _{|\alpha |\leq k}{k \choose \alpha }\|D^{\alpha }f\|^{2}\leq C\cdot \sup _{|\alpha |\leq k}|D^{\alpha }f|^{2}.}

Differential operators. D^α H_k(T²) ⊂ H_k−|α|(T²) and D^α defines a bounded linear map from H_k(T²) to H_k−|α|(T²). The operator I + Δ defines a unitary map of H_k+2(T²) onto H_k(T²); in particular (I + Δ)^k defines a unitary map of H_k(T²) onto H_−k(T²) for k ≥ 0.

The first assertions follow because D^α e_λ = λ^α e_λ and |λ^α| ≤ |λ|^|α| ≤ (1 + |λ|²)^|α|/2. The second assertions follow because I + Δ acts as multiplication by 1 + |λ|² on e_λ.

Duality. For k ≥ 0, the pairing sending f, g to (f,g) establishes a duality between H_k(T²) and H_−k(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) k f, g) (- k)

Multiplication operators. If h is a smooth function then multiplication by h defines a continuous operator on H_k(T²).

For k ≥ 0, this follows from the formula for ||f||2
(k) above and the Leibniz rule. Continuity for H_−k(T²) follows by duality, since

(f, hg) = (h f, g)

Sobolev spaces and differentiability (Sobolev's embedding theroem). For k ≥ 0, $H k +2 (T 2) \subset C k (T 2)$ and sup_|α|≤k |D^αf| ≤ C_k ⋅ ||f||_(k+2).

The inequalities for trigonometric polynomials imply the containments. The inequality for k = 0 follows from

\displaystyle {\sup |\sum a_{\lambda }e_{\lambda }|\leq \left(\sum (1+|\lambda |^{2})^{-2}\right)^{1/2}\cdot \sum |a_{\lambda }|\leq \left(\sum |a_{\lambda }|^{2}(1+|\lambda |^{2})^{2}\right)^{1/2}=\left(\sum (1+|\lambda |^{2})^{-2}\right)^{1/2}\cdot \|\sum a_{\lambda }e_{\lambda }\|_{(2)},}

by the Cauchy-Schwartz inequality. The first term is finite by the integral test, since ∬_C (1 + |z|²)⁻² dx dy =

2π \int \infty 0 (1 + r 2) -2 r dr

< ∞ using polar coordinates. In general if |α| ≤ k, then |sup D^αf| ≤ C₀ ||D^αf||₂ ≤ C₀ ⋅ C_α ⋅ ||f||_k+2 by the continuity properties of D^α.

Smooth functions. C^∞(T²) = $\cap$ H_k(T²) consists of Fourier series ∑ a_λ e_λ such that for all k > 0, (1 + |λ|²)^k |a_λ| tends to 0 as |λ| tends to ∞, i.e. the Fourier coefficients a_λ 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 H_k(T²) is a subspace of H_j(T²) and the inclusion H_k(T²) $\rightarrow$ H_j(T²) is compact.

With respect to the natural orthonormal bases, the inclusion map becomes multiplication by (1 + |λ|²)^−(k−j)/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) = $\cup$ H_k(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(ψ u) = (Δψ) u + 2(ψ x u x + ψ y u y) + ψ Δ u

, so

ψ u = (I + Δ) -1 [ψ u + (Δψ) u + 2(ψ x u x + ψ y u y) + ψ f]

. If it is known that φu lies in H_k(T²) for some k and all φ vanishing off U, then differentiating shows that φu_x and φu_y lie in H_k−1(T²). The square-bracketed expression therefore also lies in H_k−1(T²). The operator (I + Δ)⁻¹ carries this space onto H_k+1(T²), so that ψu must lie in H_k+1(T²). Continuing in this way, it follows that ψu lies in

\cap

H_k(T²) = C^∞(T²).

Hodge decomposition on functions. H₀(T²) = ∆ H₂(T²) $\oplus$ ker ∆ and C^∞(T²) = ∆ C^∞(T²) $\oplus$ 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

\oplus

ker T. This operator is bounded, self-adjoint and diagonalized by the orthonormal basis e_λ 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 e_λ = |λ|⁻²(1 + |λ|²) e_λ for λ ≠ 0. So im T must be closed and hence L²(T²) = (ker T)^⊥

\oplus

ker T = im T

\oplus

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.^[17]

Hodge theory on T². Let Ω^k(T²) be the space of smooth k-forms for 0 ≤ k ≤ 2. Thus Ω⁰(T²) = C^∞(T²), Ω¹(T²) = C^∞(T²) dx $\oplus$ C^∞(T²) dy and Ω²(T²) = C^∞(T²) dx ∧ dy. 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 dx ∧ dy and *(g dx ∧ dy) = g. Thus ** = (−1)^k on k-forms. There is a natural complex inner product on Ω^k(T²) defined by

\displaystyle {(\alpha ,\beta )=\int _{{\mathbf {T} }^{2}}\alpha \wedge \star {\overline {\beta }}.}

Define

δ = -* d *

. Thus δ takes Ω^k(T²) to Ω^k−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^[18]

\displaystyle {(d\alpha ,\beta )=\int d\alpha \wedge \star {\overline {\beta }}=\int d\alpha \wedge \star {\overline {\beta }}-\int d(\alpha \wedge \star {\overline {\beta }})=(-1)^{\partial \alpha }\int \alpha \wedge d\star {\overline {\beta }}=\int \alpha \wedge \star {\overline {\delta \beta }}=(\alpha ,\delta \beta ).}

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 dx \land dy) = (∆ f) dx \land dy

. This allows the Hodge decomposition to be generalised to include 1-forms and 2-forms:

Hodge theorem. Ω^k(T²) = ker d $\cap$ ker d∗ $\oplus$ im d $\oplus$ im ∗d = ker d $\cap$ ker d* $\oplus$ im d $\oplus$ im d*. In the Hilbert space completion of Ω^k(T²) the orthogonal complement of $\oplus$ is $\cap$ , the finite-dimensional space of harmonic k-forms, i.e. the constant k-forms. In particular in $Ω k (T 2)$ , $\cap$ , 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, Ω^k(T²) = ker ∆_k

\oplus

im ∆_k. Since ∆_k = dd* + d*d, ker ∆_k = ker d

\cap

ker d*. Moreover im (dd* + d*d) ⊊ im d

\oplus

im d*. Since ker d

\cap

ker d* is orthogonal to this direct sum, it follows that Ω^k(T²) = ker d

\cap

ker d*

\oplus

im d

\oplus

im d*. The last assertion follows because ker d contains

\cap

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

\displaystyle {(\alpha ,\beta )=\int _{X}\alpha \wedge \star {\overline {\beta }},}

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 $(df,* dg) = \int X df \land d g = \int X d (f d g) = 0$ , these are orthogonal subspaces. Let H₀ denote the orthogonal complement (H₁ $\oplus$ H₂)^⊥ = H⊥
1 $\cap$ H⊥
2.^[19]

Theorem (Hodge−Weyl decomposition). H = H₀ $\oplus$ H₁ $\oplus$ 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 ψ_n with 0 ≤ ψ_n ≤ 1 such that ∫_X ψ_n Ω tends to ∫_X Ω = ||ω||². Let

φ n = 1 - (1 - ψ n) 1/2

, a continuous function of compact support with

0 \leq φ n \leq 1

. Then ω_n = φ_n ⋅ ω tends to ω in H, since ||ω − ω_n||² = ∫_X (1 − ψ_n) Ω tends to 0.

If ω in H is such that ψ ⋅ ω is continuous for every ψ in C_c(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 C_c(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 ω_U on U. If V is another open set intersecting U, then ω_U = ω_V on U

\cap

V: in fact if z lies in U

\cap

V and ψ in C_c(U

\cap

V) ⊂ C_c(X) with ψ = 1 near z, then ψ ⋅ ω_U = ψ ⋅ ω = ψ ⋅ ω_V, so that ω_U = ω_V near z. Thus the ω_U's patch together to give a continuous 1-form ω₀ on X. By construction, ψ ⋅ ω = ψ ⋅ ω₀ for every ψ in C_c(X). In particular for φ in C_c(X) with

0 \leq φ \leq 1

, ∫ φ ⋅ ω₀ ∧ ∗ ω₀ = ||φ^1/2 ⋅ ω₀||² = ||φ^1/2 ⋅ 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 ω_n in Ω1
c(X). Take ψ_n in C_c(X) with 0 ≤ ψ_n ≤ 1 with

ψ n \cdot ω n = ω n

. Since real-valued continuous functions are closed under lattice operations. it can further be assumed that ∫ ψ2
n ω₀ ∧ ∗ ω₀, and hence ∫ ψ_n ω₀ ∧ ∗ ω₀, increase to ||ω₀||². But then ||ψ_n ⋅ ω − ω|| and ||ψ_n ⋅ ω₀ − ω₀|| tend to 0. Since

ψ n \cdot ω = ψ n \cdot ω 0

, this shows that

ω = ω 0

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) = \int X df \land *ω = -\int X f d *ω = 0

and

(* df,ω) = \int X df \land ω = - \int X f d ω = 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 D ⊂ T² 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

\oplus

H₀(T²)dy. Moreover if ψ is in C∞
c(U) then

T m (ψ) = m (ψ \circ 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 = T(ω₁), then (ω₂, dg) = 0 = (ω₂, ∗dg) for g in

C \infty c (D 1)

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 C_c(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 $\partial$ and ${\bar {\partial }}$ , it follows that

\displaystyle {dC_{c}^{\infty }(X)\oplus *dC_{c}^{\infty }(X)=\partial C_{c}^{\infty }(X)\oplus {\overline {\partial }}C_{c}^{\infty }(X),}

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 = 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.^[20]

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 $\partial$ f for f smooth.
A smooth 1-form in H₃ has the form ${\bar {\partial }}$ 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 df_n with f_n smooth of compact support. The 1-form α must be closed since, for any real-valued f in C∞
c(X),

\int _{X}f\,d\alpha =-\int _{X}df\wedge \alpha =(\alpha ,*df)=0,

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

\int _{X}\alpha \wedge \star \beta =(\alpha ,\beta )=\lim(df_{n},\beta )=\lim \int _{X}df_{n}\wedge \beta =\lim \int _{X}d(f_{n}\beta )=0.

The above characterisations have an immediate corollary:

A smooth 1-form α in H⊥
0 can be decomposed uniquely as α = da + ∗db = $\partial$ f + ${\bar {\partial }}$ 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 $\partial$ with ω harmonic, square integrable and $a, b, f, g$ smooth with square integrable differentials.^[21]

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}.^[22]

The double pole condition is invariant under holomorphic coordinate change z $\mapsto$ 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.

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 H₀ ∩ H₁ = (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

\alpha =-d\left({\psi (z) \over z}\right),

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 + da – i∗da 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 − i∗da 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

{\bar {\partial }}

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 − i∗da and the fact that ω₀ is harmonic.

Corollary of proof. ^[23] 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 H₀ ∩ H₁ = (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

(\delta ,dh)=\int _{X}\delta \wedge \star dh=\int _{Y}\delta \wedge \star dh=(\delta |_{Y},dh)_{Y}=0.

Dirichlet's principle on a Riemann surface

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.^[29]

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

\alpha =d\left(\psi (z)\log {z-a \over z-b}\right),

which locally has the form

{dz \over z-a}-{dz \over z-b}.

Poisson equation

Notes

↑ Springer 1957, p. 165
↑ Napier & Ramachandran 2011, pp. 443–444
↑ Donaldson 2011, pp. 70−71
↑
See:
- Springer 1957, pp. 151–158
- Kodaira 2007, pp. 272–275
↑ It is not assumed that if two U_i's intersect then their intersection is connected, as is the case for disks in the plane. Note, however, that if the U_i 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 U_i would be geodesically convex and hence connected; see do Carmo 1976, pp. 303–305.
↑ Kodaira 2007, pp. 290–292
↑ Kodaira 2007, pp. 290–292
↑ Kodaira 2007, pp. 251–256
↑
See:
- Weyl 1955, pp. 72–78
- Springer 1957, pp. 158–163
- Kodaira 2007, pp. 284–290
↑ Kodaira 2007, pp. 292–293
↑ Springer 1957, pp. 200–201
↑ Kodaira 2007, p. 294
↑ Springer 1957, pp. 168–172
↑
For treatments in texts on Riemann surfaces, see:
↑
For treatments in texts on partial differential equations, see for example:
↑
See:
- Hörmander 1990
- Rudin 1973, pp. 190–191
↑ 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.
↑ Warner 1983, pp. 220–221
↑ Springer 1957, pp. 178–206
↑ Springer 1957, pp. 200–201
↑ Springer 1957, pp. 195–205
↑ Springer 1957, pp. 209–211
↑ Springer 1957, pp. 209–212
↑ Springer 1957, pp. 209–212, 219
↑ Springer 1957, pp. 211–212
↑ Kodaira 2007, pp. 294−318
↑ Weyl 1955, pp. 93–118
↑ Kodaira & 312−314
↑ Springer 1957, pp. 212–213
↑ Donaldson 2011, pp. 70–71
↑ Helgason 2001, p. 442–449

References

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Differential forms on a Riemann surface

Hodge star on 1-forms

Poincaré lemma

Integration of 2-forms

Integration of 1-forms along paths

Green–Stokes formula

Duality between 1-forms and closed curves

Holomorphic and harmonic 1-forms

Sobolev spaces on T2

Hilbert space of 1-forms

Holomorphic 1-forms with a double pole

Dirichlet's principle on a Riemann surface

Holomorphic 1-forms with two single poles

Poisson equation

See also

Notes

References

Sobolev spaces on T²