On false branch points of incompressible branched immersions. (English) Zbl 0701.53009
Branched minimal immersions appear naturally as solutions of Plateau’s problem. A basic question is to decide when such an immersion is in fact unbranched. If the minimal surface is area minimizing in a three- manifold, we have a satisfactory answer from the results of Osserman, Alt and Gulliver. In this paper the authors obtain a regularity result under assumptions related to incompressibility of the surface. More precisely, given a branched minimal immersion f from a Riemann surface with boundary M into a Riemannian manifold N, if f induces an isomorphism between the fundamental groups of M and N, and is one-to-one on the boundary of M, then f has no ramified points.
Reviewer: A.Ros-Mulero
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
References:
| [1] | Alt, H. W.: Verzweigungspunkte vonH-Flächen, I. Math Z.127, 333-362 (1972); II. Math. Ann.201, 33-56 (1973) · Zbl 0253.58007 · doi:10.1007/BF01111392 |
| [2] | Elwin, J. and Short D.: Branched immersions onto compact orientable surfaces. Pac. J. Math.54, 113-122 (1974) · Zbl 0265.57004 |
| [3] | Elwin, J. and Short D.: Branched immersions between 2-manifolds of higher topological type. Pac. J. Math.58, 361-370 (1975) · Zbl 0308.57005 |
| [4] | Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Annals of Math.97, 275-305 (1973) · Zbl 0246.53053 · doi:10.2307/1970848 |
| [5] | Gulliver, R.: Branched immersions of surfaces and reduction of topological type, I. Math. Z.145, 267-288 (1975) · doi:10.1007/BF01215292 |
| [6] | Gulliver, R.: Finiteness of the ramified set for branched immersions of surfaces. Pac. J. Math.64, 153-165 (1976) · Zbl 0336.55002 |
| [7] | Gulliver, R.: Branches immersions of surfaces and reduction of topological type, II. Math. Ann.230, 25-48 (1977) · doi:10.1007/BF01420574 |
| [8] | Gulliver, R., Osserman, R., Royden, H. L.: A theorey of branched immersions of surfaces. Amer. J. Math.95, 750-812 (1973) · Zbl 0295.53002 · doi:10.2307/2373697 |
| [9] | Jost, J.: Conformal mappings and the Plateau-Douglas problem. J. reine angew. Math.359, 37-54 (1985) · Zbl 0568.49025 · doi:10.1515/crll.1985.359.37 |
| [10] | Lemaire, L.: Boundary value problems for harmonic and minimal maps of surfaces into manifolds. Ann. Scuola Norm. Sup. Pisa, IV, Vol. IX, 91-103 (1982) · Zbl 0532.58004 |
| [11] | Massey, W.: Algebraic topology: an introduction. Harcourt, Brace & World, Inc., 1967 · Zbl 0153.24901 |
| [12] | Milnor, J.: Topology from the differentiable viewpoint. The University press of Virginia, Charlottesville 1965 · Zbl 0136.20402 |
| [13] | Osserman, R.: A proof of the regularity everywhere of the classical solution to Plateau’s problem. Annals of Math.91, 550-569 (1970) · Zbl 0194.22302 · doi:10.2307/1970637 |
| [14] | Schoen, R., Yau, S. T.: Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. Math. 110 (1979), 127-142 · Zbl 0431.53051 · doi:10.2307/1971247 |
| [15] | Stoïlow, S.: Leçons sur les principes topologiques de la théorie des fonctions analytiques. Gauthier-Villars, Paris 1938 · JFM 64.0309.01 |
| [16] | Tomi, T., Tromba, A. J.: Existence theorems for minimal surfaces of nonzero genus spanning a contour. Memoirs Amer. Math. Soc., Vol.71, Nr. 382 (1988) · Zbl 0638.58004 |
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