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Doubly-connected minimal surfaces

1967; American Mathematical Society; Volume: 128; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1967-0211333-1

1088-6850

Paul M. Bailyn,

Advanced Numerical Analysis Techniques

Introduction.Summary of results.The purpose of this paper is to establish necessary conditions and sufficient conditions for two curves in three-dimensional euclidean space to bound a doubly-connected minimal surface.Loosely stated, it is shown that if the two curves are to bound a doubly-connected minimal surface then it is necessary that they not be far apart relative to their diameters and it is sufficient that they be close to each other in the sense that one be a small perturbation of the other.Results are obtained also for minimal surfaces of topological type other than doubly-connected.Necessary conditions.Soap film experiments and analysis of the classical case of minimal surfaces of revolution (see [1]) make plausible the conjecture that as the two boundary curves of a doubly-connected minimal surface are pulled apart, a position is always reached beyond which no doubly-connected minimal surface spanning the curves exists.One obtains such critical positions if the curves do not grow indefinitely in diameter and regardless of whether or not the shapes of the curves change.If the curve diameters are allowed to grow indefinitely then such a critical position may not occur as can be seen by the case of two separating coaxial circles obtained by taking sections of a minimal surface of revolution by separating planes perpendicular to the axis of the surface.In connection with this J. C. C. Nitsche [7] proves that if the Jordan curves yx andy2 bound a doubly-connected minimal surface then d(yx, y2) :£ 3/2 max (dx, d2) where d(yx, y2) is the distance between yx and y2 and dx and d2 are the diameters of yx and y2, respectively.If the curves yx and y2 lie in parallel planes, Nitsche [6] obtains a somewhat stronger result : Let yx lie in z = cx and y2 in z = c2 (cx < c2), respectively.Chose a pointpx = (xx, yx, cx)in the plane z = cx,in some sense the center of yx, and a point P2 = (x2, y2, c2) in the plane z = c2, in some sense the center of y2.Let r = c2 -cx and í¡?=[(x2-x1)2-T■(,y2->'1)2]1,2.Denote by 8X the maximal distance of the point />! from the curve yx and by 82 the maximal distance of the point p2 from the curve y2.Nitsche's theorem then states that ifyx and y2 bound a doubly-connected minimal surface then (r2+\d2)112<,8X + 82.