In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily "discretization" or "approximation" of smooth surfaces. The Gauss curvature and the mean curvature of discrete surfaces are defined which satisfy properties corresponding to the classical surface theory. We also discuss the convergence of a family of subdivided discrete surfaces of a given 3-valent discrete surface by using the Goldberg-Coxeter construction. Although discrete surfaces in general have no corresponding smooth surfaces, we may find one as the limit.
We are concerned with spectral problems of the Goldberg-Coxeter construction for 3and 4-valent finite graphs. The Goldberg-Coxeter constructions GC k,l (X) of a finite 3-or 4-valent graph X are considered as "subdivisions" of X, whose number of vertices are increasing at order O(k 2 + l 2 ), nevertheless which have bounded girth. It is shown that the first (resp. the last) o(k 2 ) eigenvalues of the combinatorial Laplacian on GC k,0 (X) tend to 0 (resp. tend to 6 or 8 in the 3-or 4-valent case, respectively) as k goes to infinity. A concrete estimate for the first several eigenvalues of GC k,l (X) by those of X is also obtained for general k and l. It is also shown that the specific values always appear as eigenvalues of GC 2k,0 (X) with large multiplicities almost independently to the structure of the initial X. In contrast, some dependency of the graph structure of X on the multiplicity of the specific values is also studied.
Abstract. In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.
The blow-up analysis for a sequence of exponentially harmonic maps from a closed surface is studied to reestablish an existence result of harmonic maps from a closed surface into a closed manifold whose 2-dimensional homotopy class vanishes.
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