Riemann surfaces, classification of

Studies of Riemann surfaces related to the behaviour of different classes of functions on these surfaces.

A complex-valued function $ f: R \rightarrow \overline{\mathbf C}\; $ on a Riemann surface $ R $ is said to be analytic on $ R $ if for any point $ p _ {0} \in R $ there exists a neighbourhood $ U $ and a local uniformizing parameter $ z = \phi ( p) $, $ \phi ( p _ {0} ) = 0 $, mapping $ U $ homeomorphically onto the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $, such that the composite function $ F( z) = f[ \phi ^ {-} 1 ( z)] $ is a single-valued analytic function on $ D $. Similarly, one can define on Riemann surfaces real-valued and complex-valued harmonic functions, subharmonic functions, etc. Let $ W $ be some conformally-invariant class of functions on the Riemann surface $ R $ containing the constants. The problem of the classification of Riemann surfaces in its simplest formulation consists in determining conditions under which a given Riemann surface $ R $ will or will not belong to the class $ {\mathcal O} _ {W} $ of those Riemann surfaces for which the class $ W $ on them consists only of the constants. The classification theory of Riemann surfaces arose in the 20th century from the classical Riemann theorem on the conformal mapping of simply-connected Riemann surfaces, the problem of types, the problem of the existence of a Green function of a Riemann surface, and the concept of the ideal boundary of a Riemann surface.

Riemann's mapping theorem states that any simply-connected Riemann surface $ R $ can be mapped conformally (and, hence, homeomorphically) onto a domain $ D $ of exactly one of the following types: $ D = \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $— the extended complex plane (the case of a Riemann surface $ R $ of elliptic type); $ D = \mathbf C $— the finite complex plane ( $ R $ is of parabolic type); or $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $— the unit disc ( $ R $ is of hyperbolic type). Since the elliptic case differs from the others already from the topological point of view, the difficult problem of recognizing whether a given Riemann surface $ R $ is of hyperbolic or parabolic type is still left. This is the classical problem of types, which is unsolved until now (1991). It is known that a closed Riemann surface of genus $ g $ for $ g= 0 $ is of elliptic type, for $ g= 1 $ it is of parabolic type, and for $ g> 1 $ of hyperbolic type; therefore, the problem of types is mainly important for open Riemann surfaces. In the case of an arbitrary, not necessarily simply-connected, Riemann surface $ R $, its type is the same as the type of its universal covering surface (see Universal covering) $ \widehat{R} $, which is always simply connected.

For simply-connected finite Riemann surfaces $ R $, the problem of finding a conformal mapping from $ R $ onto the unit disc $ D $ is equivalent to the problem of finding the Green function $ G( p, p _ {0} ) $ for $ R $, i.e. a positive harmonic function with logarithmic singularity of the form $ \mathop{\rm ln} ( 1/| z- z _ {0} |) $ at the pole $ p _ {0} \in R $( $ z = \phi ( p) $ is a parameter in a neighbourhood of $ p _ {0} $, $ z _ {0} = \phi ( p _ {0} ) $), vanishing at all points of the boundary $ \partial R $. The Green function can be also constructed for multiply-connected finite Riemann surfaces of hyperbolic type. In the case of an arbitrary open Riemann surface $ R $ one can construct an exhaustion $ \{ \overline{R}\; _ \nu \} _ {\nu = 1 } ^ \infty $ of the surface $ R $ by finite Riemann surfaces $ \overline{R}\; _ \nu $ with boundary and having Green functions

$$ G _ \nu ( p, p _ {0} ) = \mathop{\rm ln} \frac{1}{| z- z _ {0} | } + \gamma _ \nu + O(| z- z _ {0} |),\ z \rightarrow z _ {0} $$

(or $ G _ \nu ( p, p _ {0} ) \equiv + \infty $ from some index $ \nu $ onwards), and such that $ \overline{R}\; _ \nu \subset R _ {\nu + 1 } $, $ \cup _ {\nu + 1 } ^ \infty \overline{R}\; _ \nu = R $. The constant $ \gamma _ \nu $, $ - \infty < \gamma _ \nu \leq + \infty $, is called the Robin constant of the Riemann surface $ \overline{R}\; _ \nu $; $ c _ \nu = e ^ {- \gamma _ \nu } $ is the capacity of the boundary $ \partial \overline{R}\; _ \nu $( relative to the fixed pole $ p _ {0} \in R $). When $ \nu $ tends to $ \infty $, the values of $ G _ \nu ( p, p _ {0} ) $ and $ \gamma _ \nu $ can only increase. The Green function of an open Riemann surface $ R $ is defined as the limit $ G( p, p _ {0} ) $ of the increasing sequence $ \{ G _ \nu ( p, p _ {0} ) \} $ if such a limit exists; otherwise, if

$$ \lim\limits _ {\nu \rightarrow \infty } G _ \nu ( p, p _ {0} ) \equiv + \infty , $$

one says that the Riemann surface $ R $ does not have a Green function. The existence or non-existence of the Green function is independent of the choice of the pole $ p _ {0} \in R $. The class of Riemann surfaces for which the Green function does not exist is denoted by $ {\mathcal O} _ {G} $. In other words, the class $ {\mathcal O} _ {G} $ is characterized by

$$ \lim\limits _ {\nu \rightarrow \infty } \gamma _ \nu \equiv + \infty \ \textrm{ or } \ \ \lim\limits _ {\nu \rightarrow \infty } c _ \nu = 0; $$

moreover, these relations are independent of the choice of the pole as well.

Let $ R $ be an open Riemann surface and let $ \{ \Delta _ \nu \} _ {\nu = 1 } ^ \infty $ be a so-called defining sequence of closed domains $ \Delta _ \nu $ on $ R $, i.e. a sequence such that: 1) the boundary of $ \Delta _ \nu $ is a simple closed curve on $ R $; 2) $ \Delta _ {\nu + 1 } \subset \Delta _ \nu $, $ \nu = 1, 2 ,\dots $; 3) $ \cap _ {\nu = 1 } ^ \infty \Delta _ \nu = \emptyset $, i.e. the $ \Delta _ \nu $ are not compact in $ R $. Two defining sequences $ \{ \Delta _ \nu \} $ and $ \{ \Delta _ \nu ^ \prime \} $ are equivalent if to each $ \nu $ there correspond $ n $ and $ m $ such that $ \Delta _ {n} ^ \prime \subset \Delta _ \nu $ and $ \Delta _ {m} \subset \Delta _ \nu ^ \prime $. Equivalence classes of defining sequences are called boundary elements of $ R $, and the set of all boundary elements forms the ideal boundary $ \Gamma $ of $ R $, considered as a topological space. For instance, the ideal boundary of the unit disc $ D $ consists of one boundary element. Note that the Green function of an open Riemann surface $ R $, unlike the case of a hyperbolic finite Riemann surface, does not necessarily vanish on all elements of the ideal boundary $ \Gamma $. The class $ {\mathcal O} _ {G} $ is also characterized as the class of Riemann surfaces with ideal boundary of zero capacity or, for short, as the class of Riemann surfaces with zero boundary. If $ R \notin {\mathcal O} _ {G} $, then $ \lim\limits _ {\nu \rightarrow \infty } c _ \nu = c > 0 $ is called the capacity of the ideal boundary. The existence or non-existence of the Green function of a Riemann surface $ R $ and also the contents of other function classes on $ R $ are determined first of all by this and other more subtle characteristics of the ideal boundary related to the function classes themselves.

The principal function classes $ W $ on a Riemann surface $ R $ are the following:

$ \mathop{\rm AB} $— the class of bounded single-valued analytic functions on $ R $;

$ \mathop{\rm AD} $— the class of single-valued analytic functions $ f( z) $ with a finite Dirichlet integral on $ R $:

$$ D _ {R} ( f ) = {\int\limits \int\limits } _ { R } \left | \frac{dw}{dz} \right | ^ {2} dx dy,\ \ z = x + iy; $$

$ \mathop{\rm HP} $, $ \mathop{\rm HB} $ and $ \mathop{\rm HD} $— the classes of single-valued harmonic functions on $ R $ with, respectively, a positive, a bounded and a finite Dirichlet integral. These classes can be combined; for example, $ \mathop{\rm ABD} $ is the class of bounded single-valued analytic functions with a finite Dirichlet integral on $ R $. For the corresponding classes $ {\mathcal O} _ {W} $ of $ R $ the following strict inclusions and equalities have been established:

$$ {\mathcal O} _ {G} \subset {\mathcal O} _ \mathop{\rm HP} \subset {\mathcal O} _ \mathop{\rm HB} \subset {\mathcal O} _ \mathop{\rm AB} \subset {\mathcal O} _ \mathop{\rm ABD} = {\mathcal O} _ \mathop{\rm AD} , $$

$$ {\mathcal O} _ \mathop{\rm HB} \subset {\mathcal O} _ \mathop{\rm HD} \subset {\mathcal O} _ \mathop{\rm AD} ,\ {\mathcal O} _ \mathop{\rm HBD} = {\mathcal O} _ \mathop{\rm HD} . $$

For domains $ R $ in the plane, these relations can be simplified:

$$ {\mathcal O} _ {G} = {\mathcal O} _ \mathop{\rm HP} = {\mathcal O} _ \mathop{\rm HB} = {\mathcal O} _ \mathop{\rm HD} , $$

$$ {\mathcal O} _ \mathop{\rm AB} \subset {\mathcal O} _ \mathop{\rm ABD} = {\mathcal O} _ \mathop{\rm AD} . $$

Of great importance are also the Hardy classes $ \mathop{\rm AH} _ {p} $, $ 0 < p \leq + \infty $, of single-valued analytic functions $ w = f( z) $ on $ R $. For $ 0 < p < + \infty $, a function $ f \in \mathop{\rm AH} _ {p} $ if the subharmonic function $ | f | ^ {p} $ has a harmonic majorant on the entire Riemann surface $ R $, and $ \mathop{\rm AH} _ \infty = \mathop{\rm AB} $( see Boundary properties of analytic functions).

A Riemann surface of parabolic type is of class $ {\mathcal O} _ {G} $, therefore the problem of characterizing Riemann surfaces of class $ {\mathcal O} _ {G} $ is sometimes called the generalized problem of types. There are many results in which the condition that a Riemann surface belongs to the above-mentioned classes is established in different terms. Deep studies were dedicated to finding out the intrinsic properties of Riemann surfaces of the given classes. In particular, it turned out that Riemann surfaces with zero boundary are in many respects analogous to closed Riemann surfaces. The analogues of Abelian differentials (cf. Abelian differential) and the corresponding integrals can be constructed on them.

More subtle properties of the ideal boundary of a Riemann surface $ R $ can be studied also by different compactifications of $ R $. For example, let $ N( R) $ be the Wiener algebra of functions $ u $ on a Riemann surface $ R $ that are bounded, continuous and harmonizable on $ R $; the latter means that for any regular domain $ G \subset R $ there exists a generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see Perron method) with boundary values $ u $ on the boundary $ \partial G $. The Wiener compactification of $ R $ is the compact Hausdorff space $ R ^ \star $ such that $ R $ is an open dense subspace of $ R ^ \star $, each function $ u \in N( R) $ can be continuously continued onto $ R ^ \star $ and $ N( R) $ separates the points of $ R ^ \star $. The Wiener compactification exists for any Riemann surface $ R $. The set $ \Gamma ( R) = R ^ \star \setminus R $ is called the Wiener ideal boundary of $ R $, and the subset $ \Delta ( R) \subset \Gamma ( R) $ of those points of $ R ^ \star $ at which all potentials from $ N( R) $ vanish, is called the Wiener harmonic boundary. In these terms, for example, the inclusion $ R \in {\mathcal O} _ {G} $ is equivalent to the equality $ \Delta ( R) = \emptyset $; from this the strict inclusion $ {\mathcal O} _ \mathop{\rm HP} \subset {\mathcal O} _ \mathop{\rm HB} $ follows also.

The problem of removable sets (cf. Removable set) on Riemann surfaces is also related to the classification of Riemann surfaces. Thus, a compactum $ K $ on a Riemann surface $ R $ is called $ \mathop{\rm AB} $- removable if for some neighbourhood $ U \supset K $ on $ R $ all $ \mathop{\rm AB} $- functions on $ U \setminus K $ have an analytic continuation to the entire neighbourhood $ U $.

The attention of many scientists was also attracted by the problems of the classification of Riemannian manifolds of arbitrary dimensions $ N \geq 2 $, related to the studies of the classes of functions described above.

For references see Riemann surface.

Comments

In addition to the references quoted in Riemann surface and Riemann surfaces, conformal classes of, see also [a1]–[a2].

References

[a1]	M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
[a2]	C. Constantinescu, A. Cornea, "Ideale Ränder Riemannscher Flächen" , Springer pp. 1963