Geodesic nets on the \(2\)-sphere. (English) Zbl 0871.53038

Hass, Joel; Morgan, Frank

Proc. Am. Math. Soc. 124, No. 12, 3843-3850 (1996).

A geodesic net is a graph embedded on a surface whose edges are geodesic arcs and whose vertices consist of arcs whose unit tangent vector sum is zero at each vertex. The aim of this paper is to study geodesic nets on a compact surface. The main result of this work can be stated as follows: Let \(S\) be a 2-sphere with a smooth Riemannian metric with positive curvature. There exists a geodesic net \(G\) partitioning \(S\) into three components.

Reviewer: G.Tsagas (Thessaloniki)

MSC:

53C22	Geodesics in global differential geometry
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

geodesic net; 2-sphere

References:

[1]	F. J. Almgren, Jr., and Jean E. Taylor, The geometry of soap films and soap bubbles, Scientific American, July, 1976, 82-93.
[2]	Jaigyoung Choe, On the existence and regularity of fundamental domains with least boundary area, J. Differential Geom. 29 (1989), no. 3, 623 – 663. · Zbl 0643.53041
[3]	Christopher B. Croke, Poincaré’s problem and the length of the shortest closed geodesic on a convex hypersurface, J. Differential Geom. 17 (1982), no. 4, 595 – 634 (1983). · Zbl 0501.53031
[4]	Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71 – 111. · Zbl 0686.53036 · doi:10.2307/1971486
[5]	J. Hass and F. Morgan, Geodesics and soap bubbles on surfaces (Math. Z., to appear). · Zbl 0865.53009
[6]	A. Heppes, Isogonale sphärische Netze, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 7 (1964), 41 – 48 (German). · Zbl 0127.37601
[7]	H. Howards, Soap bubbles on surfaces, undergraduate thesis, Williams College, 1992.
[8]	L. Lusternik and L. Schnirelman, Sur le probleme de trois géodesiques fermées sur les surface de genre 0, C. R. Acad. Sci. Paris 189 (1929), 269-271. · JFM 55.0316.02
[9]	Frank Morgan, Soap bubbles in \?² and in surfaces, Pacific J. Math. 165 (1994), no. 2, 347 – 361. · Zbl 0820.53002
[10]	Frank Morgan, Size-minimizing rectifiable currents, Invent. Math. 96 (1989), no. 2, 333 – 348. · Zbl 0645.49024 · doi:10.1007/BF01393966
[11]	Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489 – 539. , https://doi.org/10.2307/1970949 Jean E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in \?³, Ann. of Math. (2) 103 (1976), no. 3, 541 – 546. · Zbl 0335.49033 · doi:10.2307/1970950