# 2-Dimension from the Topological Viewpoint by Barmak J.A., Minian E.G.

By Barmak J.A., Minian E.G.

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Induccion en la Geometria

L. a. INDUCCION EN GEOMETRIA de I. L. GOLOVINA

Geometry and Spectra of Compact Riemann Surfaces

This vintage monograph is a self-contained advent to the geometry of Riemann surfaces of continuous curvature –1 and their size and eigenvalue spectra. It makes a speciality of topics: the geometric thought of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces.

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Discrete S-isothermic surfaces are therefore composed of tangent spheres and tangent circles, with the spheres and circles intersecting orthogonally. The class of discrete S-isothermic surfaces is obviously invariant under M¨obius transformations. D/ ! R3 . The discrete isothermic surface obtained is called the central extension of the discrete S-isothermic surface. All its faces are orthogonal kites. An important fact of the theory of isothermic surfaces (smooth and discrete) is the existence of a dual isothermic surface [6].

The construction of discrete S-isothermic “round spheres” is based on their relation to circle packings in S 2 . The following theorem is central in this theory. 1. For every polytopal1 cellular decomposition of the sphere, there exists a pattern of circles in the sphere with the following properties. There is a circle corresponding to each face and to each vertex. The vertex circles form a packing with two circles touching if and only if the corresponding vertices are adjacent. Likewise, the face circles form a packing with circles touching if and only if the corresponding faces are 1 We call a cellular decomposition of a surface polytopal if the closed cells are closed discs, and two closed cells intersect in one closed cell if at all.

1 (Christoffel). , a part of the unit sphere. Taking this as a deﬁnition for smooth minimal surfaces and discretizing all notions in a convenient way, we are led to the following deﬁnition suggested in [2]. 2. A discrete minimal surface is deﬁned to be a discrete S-isothermic surface (made of touching spheres and orthogonal intersecting circles; see [1, 2] for more details) such that all spheres of the dual discrete S-isothermic surface intersect one ﬁxed additional sphere orthogonally. This ﬁxed sphere is taken to be the unit sphere S 2 .