View source for Riemann surfaces, classification of

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

A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820501.png" /> on a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820502.png" /> is said to be analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820503.png" /> if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820504.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820505.png" /> and a [[Local uniformizing parameter|local uniformizing parameter]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820507.png" />, mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820508.png" /> homeomorphically onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r0820509.png" />, such that the composite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205010.png" /> is a single-valued [[Analytic function|analytic function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205011.png" />. Similarly, one can define on Riemann surfaces real-valued and complex-valued harmonic functions, subharmonic functions, etc. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205012.png" /> be some conformally-invariant class of functions on the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205014.png" /> will or will not belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205015.png" /> of those Riemann surfaces for which the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205016.png" /> 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205017.png" /> can be mapped conformally (and, hence, homeomorphically) onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205018.png" /> of exactly one of the following types: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205019.png" /> — the extended complex plane (the case of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205020.png" /> of elliptic type); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205021.png" /> — the finite complex plane (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205022.png" /> is of parabolic type); or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205023.png" /> — the unit disc (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205025.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205027.png" /> is of elliptic type, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205028.png" /> it is of parabolic type, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205029.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205030.png" />, its type is the same as the type of its universal covering surface (see [[Universal covering|Universal covering]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205031.png" />, which is always simply connected.

For simply-connected finite Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205032.png" />, the problem of finding a conformal mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205033.png" /> onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205034.png" /> is equivalent to the problem of finding the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205036.png" />, i.e. a positive harmonic function with logarithmic singularity of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205037.png" /> at the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205038.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205039.png" /> is a parameter in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205041.png" />), vanishing at all points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205042.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205043.png" /> one can construct an exhaustion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205044.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205045.png" /> by finite Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205046.png" /> with boundary and having Green functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205047.png" /></td> </tr></table>

(or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205048.png" /> from some index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205049.png" /> onwards), and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205051.png" />. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205053.png" />, is called the [[Robin constant|Robin constant]] of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205054.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205055.png" /> is the [[Capacity|capacity]] of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205056.png" /> (relative to the fixed pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205057.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205058.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205059.png" />, the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205061.png" /> can only increase. The Green function of an open Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205062.png" /> is defined as the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205063.png" /> of the increasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205064.png" /> if such a limit exists; otherwise, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205065.png" /></td> </tr></table>

one says that the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205066.png" /> does not have a Green function. The existence or non-existence of the Green function is independent of the choice of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205067.png" />. The class of Riemann surfaces for which the Green function does not exist is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205068.png" />. In other words, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205069.png" /> is characterized by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205070.png" /></td> </tr></table>

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205071.png" /> be an open Riemann surface and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205072.png" /> be a so-called defining sequence of closed domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205074.png" />, i.e. a sequence such that: 1) the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205075.png" /> is a simple closed curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205076.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205078.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205079.png" />, i.e. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205080.png" /> are not compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205081.png" />. Two defining sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205083.png" /> are equivalent if to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205084.png" /> there correspond <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205086.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205088.png" />. Equivalence classes of defining sequences are called boundary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205089.png" />, and the set of all boundary elements forms the ideal boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205091.png" />, considered as a topological space. For instance, the ideal boundary of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205092.png" /> consists of one boundary element. Note that the Green function of an open Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205093.png" />, unlike the case of a hyperbolic finite Riemann surface, does not necessarily vanish on all elements of the ideal boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205094.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205095.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205096.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205097.png" /> is called the capacity of the ideal boundary. The existence or non-existence of the Green function of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205098.png" /> and also the contents of other function classes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r08205099.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050100.png" /> on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050101.png" /> are the following:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050102.png" /> — the class of bounded single-valued analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050103.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050104.png" /> — the class of single-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050105.png" /> with a finite [[Dirichlet integral|Dirichlet integral]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050106.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050107.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050110.png" /> — the classes of single-valued harmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050111.png" /> with, respectively, a positive, a bounded and a finite Dirichlet integral. These classes can be combined; for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050112.png" /> is the class of bounded single-valued analytic functions with a finite Dirichlet integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050113.png" />. For the corresponding classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050114.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050115.png" /> the following strict inclusions and equalities have been established:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050116.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050117.png" /></td> </tr></table>

For domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050118.png" /> in the plane, these relations can be simplified:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050119.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050120.png" /></td> </tr></table>

Of great importance are also the [[Hardy classes|Hardy classes]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050122.png" />, of single-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050123.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050124.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050125.png" />, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050126.png" /> if the subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050127.png" /> has a harmonic majorant on the entire Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050128.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050129.png" /> (see [[Boundary properties of analytic functions|Boundary properties of analytic functions]]).

A Riemann surface of parabolic type is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050130.png" />, therefore the problem of characterizing Riemann surfaces of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050131.png" /> 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|Abelian differential]]) and the corresponding integrals can be constructed on them.

More subtle properties of the ideal boundary of a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050132.png" /> can be studied also by different compactifications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050133.png" />. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050134.png" /> be the Wiener algebra of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050135.png" /> on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050136.png" /> that are bounded, continuous and harmonizable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050137.png" />; the latter means that for any regular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050138.png" /> there exists a generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see [[Perron method|Perron method]]) with boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050139.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050140.png" />. The Wiener compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050141.png" /> is the compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050142.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050143.png" /> is an open dense subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050144.png" />, each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050145.png" /> can be continuously continued onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050146.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050147.png" /> separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050148.png" />. The Wiener compactification exists for any Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050149.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050150.png" /> is called the Wiener ideal boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050151.png" />, and the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050152.png" /> of those points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050153.png" /> at which all potentials from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050154.png" /> vanish, is called the Wiener harmonic boundary. In these terms, for example, the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050155.png" /> is equivalent to the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050156.png" />; from this the strict inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050157.png" /> follows also.

The problem of removable sets (cf. [[Removable set|Removable set]]) on Riemann surfaces is also related to the classification of Riemann surfaces. Thus, a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050158.png" /> on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050159.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050161.png" />-removable if for some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050162.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050163.png" /> all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050164.png" />-functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050165.png" /> have an analytic continuation to the entire neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050166.png" />.

The attention of many scientists was also attracted by the problems of the classification of Riemannian manifolds of arbitrary dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082050/r082050167.png" />, related to the studies of the classes of functions described above.

For references see [[Riemann surface|Riemann surface]].



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Ideale Ränder Riemannscher Flächen" , Springer  pp. 1963</TD></TR></table>