Differential on a Riemann surface

A differential form on a Riemann surface $ S $ that is invariant with respect to a conformal transformation of the local uniformizing parameter $ z = x + iy $. Differentials of the first order are most often encountered; these are differential forms of dimension 1 that are linear with respect to the differential of each of the variables $ ( x , y ) $, of the form

$$ \omega = p dx + q dy \equiv p ( x , y ) dx + q ( x , y ) dy , $$

$$ \pi = r dx + s dy , $$

and that are invariant with respect to parameter change with sufficiently smooth coefficients $ p , q ,\dots $. Differentials of order zero are sufficiently smooth complex functions $ f = f ( x , y ) $, $ g = g ( x , y ) $ that are invariant with respect to parameter change, i.e. functions of the points $ P \in S $; differentials of the second order have the form

$$ \Omega = A dx dy \equiv A ( x , y ) dx dy ,\ \Pi = B dx dy . $$

All differentials of order $ k > 2 $ on a Riemann surface vanish identically.

The addition of differentials of the same order on a Riemann surface is effected in the usual way:

$$ f + g ,\ \omega + \pi = ( p + r ) dx + ( q + s ) dy , $$

$$ \Omega + \Pi = ( A + B ) dx dy , $$

and is commutative and associative. The exterior multiplication of differentials on a Riemann surface is distributive with respect to addition, is denoted by the symbol $ \wedge $ and is defined by the following laws:

$$ f \wedge g = fg ; \ f \wedge \omega = ( fp) dx + ( fg) dy , $$

$$ dx \wedge dx = dy \wedge dy = 0 ,\ dy \wedge dx = - dx \wedge dy = - dx dy . $$

Hence

$$ \omega \wedge \pi = ( ps - qr) dx dy ,\ \pi \wedge \omega = - \omega \wedge \pi ; $$

$$ f \wedge \Omega = ( fA) dx dy . $$

In general, exterior multiplication of a differential of order $ k $ by a differential of order $ l $, where $ k + l \leq 2 $, yields a differential or order $ k + l $, while if $ k + l > 2 $ it vanishes. The linear differentiation operator $ d = ( \partial / \partial x ) dx + ( \partial / \partial y ) dy $ transforms a differential of order $ k $ to a differential of order $ k + 1 $:

$$ df = \frac{\partial f }{\partial x } dx + \frac{\partial f }{\partial y } dy , $$

$$ d \omega = ( dp) dx + ( dq) dy = \left ( \frac{\partial q }{\partial x } - \frac{\partial p }{\partial y } \right ) dx dy ,\ d \Omega = 0 . $$

Moreover,

$$ d ( fg) = f dg + g df , $$

$$ d ( f \omega ) = ( df ) \omega + f ( d \omega ) = - \omega ( df ) + f ( d \omega ) $$

and $ dd = 0 $. Of importance to differentials on a Riemann surface is also the linear star conjugation operator

$$ {} ^ \star \omega = - q dx + p dy . $$

Here

$$ {} ^ \star ( f \wedge \omega ) = f \wedge ( {} ^ \star \omega ) , $$

$$ \omega \wedge ( {} ^ \star \pi ) = \pi \wedge ( {} ^ \star \omega ) = ( pr + qs) dx dy , $$

$$ \omega \wedge ( {} ^ \star \omega ) = ( p ^ {2} + q ^ {2} ) dx dy ,\ {} ^ \star d = - \frac \partial {\partial y } dx + \frac \partial {\partial x } dy , $$

$$ {} ^ \star ( df ) = ( {} ^ \star d ) f ,\ ( {} ^ \star d ) \omega = {} ^ \star d \omega = - d {} ^ \star \omega . $$

The star conjugation operator is not identical with the complex conjugation operator. The latter is denoted by a bar: if $ f = g + ih $, then $ \overline{f}\; = g - ih $, $ \overline \omega \; = \overline{p}\; dx + \overline{q}\; dy $, $ \overline \Omega \; = \overline{A}\; dx dy $; also $ d \overline \omega \; = \overline{ {d \omega }}\; $, $ {} ^ \star \overline \omega \; = \overline{ {{} ^ \star \omega }}\; $. The Laplace operator $ \Delta = d {} ^ \star d $ is defined on differentials of order zero:

$$ \Delta f = d {} ^ \star df = \left ( \frac{\partial ^ {2} f }{\partial x ^ {2} } + \frac{\partial ^ {2} f }{\partial y ^ {2} } \right ) dx dy . $$

A differential $ \omega $ is called exact if there exists a function $ f $ on $ S $ such that $ \omega = df $ everywhere; if $ \omega = {} ^ \star df $, $ \omega $ is called co-exact; if $ d \omega = 0 $ on $ S $, $ \omega $ is called closed; and if $ d {} ^ \star \omega = 0 $, $ \omega $ is called a co-closed differential. Exactness entails closedness, but the opposite is not true. Let $ c _ {1} , c _ {2} \dots $ be cycles on $ S $. The integrals $ \int _ {c _ {1} } \omega , \int _ {c _ {2} } \omega \dots $ known as the periods of the differential $ \omega $, are defined in the usual way with the aid of a local uniformizing parameter. If $ c _ {1} $ and $ c _ {2} $ are homologous on $ S $ and $ \omega $ is a closed differential, then $ \int _ {c _ {1} } \omega = \int _ {c _ {2} } \omega $, i.e. the periods of a closed differential depend only on the homology classes. All periods of an exact differential are equal to zero. Conversely, a closed differential is exact if and only if all its periods are equal to zero.

A function $ f \in C ^ {2} $ is said to be harmonic on $ S $ if $ \Delta f = 0 $. A differential $ \omega \in C ^ {1} $ is said to be a harmonic differential on $ S $ if $ \omega $ is closed and co-closed: $ d \omega = d {} ^ \star \omega = 0 $. A harmonic differential $ \omega $ is the total differential of a harmonic function in a neighbourhood of each point of $ S $. If two real-valued functions $ u , v \in C ^ {1} $ on $ S $ are connected by the relation $ dv = {} ^ \star du $, they are conjugate harmonic functions satisfying the Cauchy–Riemann equations. Consequently, a function $ f = u + iv \in C ^ {1} $ is regular analytic, or holomorphic, on $ S $ if $ {} ^ \star df = - i df $. A differential $ \omega \in C ^ {1} $ is said to be a regular analytic, or holomorphic, differential on $ S $ if $ d \omega = 0 $ and $ {} ^ \star \omega = - i \omega $. A holomorphic differential $ \omega $ is the total differential of a holomorphic function in a neighbourhood of each point of $ S $. A holomorphic differential $ \omega $ can be locally represented as $ \omega = f dz $, where $ dz = dx + i dy $ and $ f $ is a holomorphic function in $ z $.

The equivalence classes of measurable complex differentials on a Riemann surface for which the integral $ \int \int _ {S} \omega \wedge ( {} ^ \star \overline \omega \; ) $ is finite form a Hilbert space $ L _ {2} ( S) $ with the usual addition, multiplication by complex scalars and scalar product $ ( \omega , \pi ) = \int \int _ {S} \omega \wedge ( {} ^ \star \overline \pi \; ) $. Each differential $ \omega $ from the class $ L _ {2} ( S) \cap C ^ {3} ( S) $ is uniquely representable as the sum $ \omega = \omega _ {h} + df + {} ^ \star dg $ where $ f , g \in C ^ {2} ( S) $ and $ \omega _ {h} $ is a harmonic differential on the Riemann surface.

The above harmonic and holomorphic functions or differentials of class $ C ^ {1} $ on $ S $ are said to be regular on $ S $. Let a differential $ \theta $ be defined, for example in a deleted neighbourhood $ U $ of a point $ P _ {0} \in S $, which is harmonic in $ U $. One then says that the harmonic differential $ \omega $ has singularity $ \theta $ in $ P _ {0} $ if the difference $ \omega - \theta $ is a regular harmonic differential.

Similar definitions are utilized for harmonic and analytic functions, analytic differentials, etc. In particular, in the case of an analytic differential $ \omega = f dz $ it is usually assumed that the function $ f $ is either regular analytic in a neighbourhood of each point $ P _ {0} \in S $ or else has only isolated singular points on $ S $ of single-valued character. An analytic differential $ \omega $ on $ S $ which has only singularities of pole type,

$$ \theta = ( a _ {-} n z ^ {-} n + a _ {-} n+ 1 z ^ {-} n+ 1 + \dots + a _ {-} 1 z ^ {-} 1 ) dz , $$

is called a meromorphic differential; here $ a _ {-} n \neq 0 $ and $ n $ is the order of the pole; if $ n = 1 $, the pole is called simple: $ a _ {-} 1 $ is called the residue of the differential $ \omega $ at the pole $ P _ {0} $. A meromorphic differential on a compact Riemann surface $ S $ is also called an Abelian differential. A harmonic function on $ S $ or on some domain $ D \subset S $ with given singularities is also known as an Abelian potential.

The integration of Abelian differentials yields Abelian integrals (cf. Abelian integral), which actually account for all integrals of algebraic functions. In studying analytic differentials on an arbitrary, usually non-compact, Riemann surface $ S $, the natural requirement of preservation of the basic features of the classical theory of differentials on compact Riemann surfaces results in imposing additional conformally invariant restrictions on the regular differentials under study. The restriction most often used is the condition of integrability of, for example, the square of an analytic differential $ \omega = f dz $, i.e. the Dirichlet integral

$$ {\int\limits \int\limits } _ { S } | f | ^ {2} d x d y $$

must be finite.

Of fundamental importance in the theory of differentials on Riemann surfaces is the problem of the existence of a harmonic and analytic differential with given singularities on an arbitrary Riemann surface $ S $. This problem is directly connected with the global uniformization problem for Riemann surfaces, since the construction of a global uniformizing parameter requires the ability to construct a differential with given singularities.

The principal results concerning the existence problem are given below.

If a cycle $ c $ not homologous to zero exists on $ S $, there also exists on $ S $ an everywhere-regular harmonic differential $ \omega $ with period $ \int _ {c} \omega \neq 0 $ and an everywhere-regular differential $ \omega + i {} ^ \star \omega $. These differentials are not exact, and for this reason their integration does not yield single-valued harmonic or analytic functions on $ S $. On a compact Riemann surface all harmonic and exact differentials are identically equal to zero. On the contrary, on a non-compact Riemann surface there exist everywhere-regular exact harmonic and holomorphic differentials which are not identically equal to zero.

Let $ P _ {0} $ be a fixed point of an arbitrary Riemann surface $ S $ and let $ n $ be an arbitrary natural number. Then there exist: an exact harmonic differential with singularity $ d ( 1 / z ^ {n} ) $ at $ P _ {0} $; an exact real harmonic differential with singularity $ \mathop{\rm Re} d ( 1 / z ^ {n} ) $( or $ \mathop{\rm Im} d ( 1 / z ^ {n} ) $) at $ P _ {0} $; a harmonic function with singularity $ 1 / z ^ {n} $ at $ P _ {0} $; and an analytic differential with singularity $ d ( 1 / z ^ {n} ) $ at $ P _ {0} $ whose real part is an exact differential.

Let $ P _ {0} $ and $ P _ {1} $ be distinct points of $ S $. Then there exist: a harmonic or an analytic differential on $ S $ with singularities $ - dz / z $ at $ P _ {0} $ and $ dz / z $ at $ P _ {1} $; and a real harmonic function on $ S $ with singularities $ - \mathop{\rm ln} | z | $ at $ P _ {0} $ and $ \mathop{\rm ln} | z | $ at $ P _ {1} $.

Let $ P _ {0} \dots P _ {n} $ be any pairwise distinct points on $ S $ and let $ c _ {0} \dots c _ {n} $ be arbitrary non-zero complex numbers such that $ c _ {0} + \dots + c _ {n} = 0 $. There exists a harmonic or an analytic differential on $ S $ which is everywhere regular except at $ P _ {0} \dots P _ {n} $, while at the points $ P _ {j} $ it has simple poles with corresponding residues $ c _ {j} $, $ j = 0 \dots n $.

The solution of the Dirichlet problem is also possible for Jordan domains $ D \subset S $, i.e. domains whose boundary $ \partial D $ consists of $ n $ non-intersecting Jordan curves $ C _ {1} \dots C _ {n} $.

The theory of differentials is the most advanced for compact Riemann surfaces $ S $. Let the genus (cf. Genus of a surface) of $ S $ be $ g $. The vector space $ \mathfrak H $ of regular harmonic differentials $ \omega $ over the field of complex numbers has dimension $ 2g $. If $ a _ {1} b _ {1} \dots a _ {g} b _ {g} $ are the cycles of a canonical basis of the homology of $ S $, one may choose as a canonical basis of the differentials in $ \mathfrak H $ the differentials $ \omega _ {i} $, $ i = 1 \dots g $, with period one along $ a _ {i} $ and with period zero along $ a _ {j} $, $ j \neq i $, and along all $ b _ {i} $; furthermore, $ \omega _ {g+} i $, $ i = 1 \dots g $, have period one along $ b _ {i} $ and period zero along $ b _ {j} $, $ j \neq i $, and along all $ a _ {i} $. Any harmonic differential $ \omega $ can be represented in the form of a linear combination

$$ \omega = \sum _ {i = 1 } ^ { g } A _ {i} \omega _ {i} + \sum _ {i = 1 } ^ { g } B _ {i} \omega _ {g + i } , $$

where $ A _ {i} $ are the so-called $ A $- periods of $ \omega $ along the cycles $ a _ {i} $, and $ B _ {i} $ are the $ B $- periods of $ \omega $ along the cycles $ b _ {i} $.

Holomorphic differentials on a compact Riemann surface are called Abelian differentials of the first kind. The dimension of the vector space of holomorphic differentials is $ g $. All the differentials on a Riemann surface which have just been discussed can be expressed in terms of the variables $ z $ and $ \overline{z}\; $, e.g.

$$ f = f ( z , \overline{z}\; ) ,\ \omega = p dz + q d \overline{z}\; = \ p ( z , \overline{z}\; ) dz + q ( z , \overline{z}\; ) d \overline{z}\; , $$

$$ \Omega = A dz \wedge d \overline{z}\; = A ( z , \overline{z}\; ) dz \wedge d \overline{z}\; , $$

etc. As distinct from complex manifolds of higher dimension, on Riemann surfaces only the exterior differential forms of types $ ( 0 , 0 ) $, $ ( 1 , 0 ) $, $ ( 0 , 1 ) $, and $ ( 1 , 1 ) $, having respective form $ f $, $ p dz $, $ q d \overline{z}\; $, $ A dz \wedge d \overline{z}\; $, are non-trivial. An analytic differential depends only on $ z $:

$$ f = f ( z) ,\ \omega = p ( z) dz . $$

Non-linear differential forms of the type $ p ( z , \overline{z}\; ) dz ^ {m} d {\overline{z}\; } {} ^ {n} $, where $ m $ and $ n $ are integers, are also employed. They are also known as differentials of type $ ( m , n ) $ or of dimension $ ( m , n ) $. Differentials of type $ ( 0 , 0 ) $ are known as functions, those of type $ ( 1 , 0 ) $ as linear differentials, those of type $ ( - 1 , 0 ) $ as inverse differentials, and those of type $ ( 2 , 0 ) $ as quadratic differentials. Quadratic differentials are the most frequently used. See also Global structure of trajectories; Local structure of trajectories; Quadratic differential; Riemann surface; Uniformization.

References

[1]	G. Springer, "Introduction to Riemann surfaces" , Chelsea, reprint (1981) MR0122987 MR1530201 MR0092855 Zbl 0501.30039
[2]	R. Nevanlinna, "Uniformisierung" , Springer (1953) MR0057335 Zbl 0053.05003
[3]	M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) MR0065652 Zbl 0059.06901
[4]	H. Behnke, F. Sommer, "Theorie der analytischen Funktionen einer komplexen Veränderlichen" , Springer (1955) MR0073682 Zbl 0065.06102

Comments

Differentials on a Riemann surface are simply complex-valued differential forms on the underlying $ 2 $- dimensional manifold (cf. also Differential form). Their local expressions in terms of local coordinates, however, are (usually) only written down with respect to local coordinates $ x , y $ such that $ z = x + i y $ is a local uniformizing parameter (i.e. $ z $ is a local coordinate on the complex manifold). This is the meaning of the phrase "invariant with respect to conformal transformations of the local uniformizing parameter" and other phrases with "invariant" in them in the article above. The definition of the $ {} ^ \star $- operator only makes sense for local expressions $ p d x + g d y $ for a differential form $ \omega $ such that $ x + i y = z $ is a local uniformizing parameter (because the conformal coordinate changes satisfy the Cauchy–Riemann equations). The differential $ {} ^ \star \omega $ is also called the conjugate differential of $ \omega $.

A pure differential is one for which $ {} ^ \star \omega = - i \omega $. This means it can be written as $ \omega = f d z $( for some (not necessarily holomorphic) function $ f $).

References

[a1]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) MR0583745 Zbl 0475.30001
[a2]	L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1960) pp. Chapt. V MR0124486 MR0114911 Zbl 0196.33801 Zbl 0100.28903