# Cut

*in a domain $ D \subset \mathbf C $*
along a non-closed simple arc $ \gamma = \{ {z ( t) } : {0 \leq t \leq 1 } \} $

The removal of the points of the arc $ \gamma $ from the domain $ D $, that is, replacing the domain $ D $ by the domain (or domains) $ D \setminus \gamma $, as well as the set $ \gamma $ itself. Here it is assumed that either the whole arc $ \gamma $ or the whole arc except the initial or end point $ z ( 0), z ( 1) $ belongs to $ D $ and that $ z ( 0) $ or $ z ( 1) $ belong to the boundary $ \partial D $. To each point $ z ( t) $ of the cut $ \gamma $, when $ 0 < t < 1 $, there correspond two prime ends of the part of the domain $ D $ which belongs to $ \gamma $; the left and right prime ends (cf. Limit elements). The union of these prime ends form the left and right-hand sides of the cut $ \gamma $.

#### Comments

A cut is also called a slit.

One also speaks of boundary elements instead of limit elements or prime ends. These notions do not coincide in general, but for "nice" domains $ D $( e.g. with Jordan boundary) they can be identified. A related notion is that of a crosscut: an open simple arc that begins and ends at two different points of $ \partial D $, cf. [a1], especially Chapt. 3.

The word "cut" also occurs in several more meanings in various parts of mathematics. Thus, there is the notion of a Dedekind cut in the real or rational numbers, cf. Real number. The notion cut point on a geodesic line; and a cut or cutset in a graph or transportation network (cf. Flow in a network for the latter). Finally there are the cut locus and the cutting problem.

#### References

[a1] | M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold (1970) |

**How to Cite This Entry:**

Cut.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cut&oldid=46567