Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities
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Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities Jacob Bernstein & Francesco Maggi Communicated by F. Lin
Abstract By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.
1. Introduction 1.1. Overview Minimal surfaces in R3 provide the standard mathematical model of soap films at equilibrium. Nevertheless, there is a historical mismatch between the classical theory of minimal surfaces, which focuses on smooth immersions with vanishing mean curvature, and the richer structures documented experimentally since the pioneering work of Plateau [26]. Indeed, two types of singular points are observed in soap films, called Y and T points; see Fig. 2 below. We call the surfaces described in experiments minimal Plateau surfaces and ask: To what extent may the classical theory of minimal surfaces be generalized to minimal Plateau surfaces and what new conclusions may be drawn? This paper studies this question in the model case provided by Schoen’s rigidity theorem for catenoids [34], a (classical) minimal surface in R3 spanning two parallel circles with centers on the same axis has rotational symmetry about this axis and so is either a pair of flat disks or a subset of a catenoid. Schoen’s theorem is an interesting model case for two reasons: (i) its extension to minimal Plateau surfaces requires the inclusion of new cases of rigidity, given by singular catenoids; (ii) Schoen’s proof uses Alexandrov’s method of moving planes [2], which has JB was partially supported by the NSF Grant DMS-1609340 and DMS-1904674. FM was partially supported by the NSF Grants DMS-156535 and DMS-FRG-1854344
J. Bernstein, F. Maggi
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Fig. 1. Two parallel circles with same radii lying at a sufficiently small distance span exactly three smooth minimal surfaces: a pair of disks, a “fat” catenoid (which is stable) and a “skinny” catenoid (which is unstable). The same circles span five minimal Plateau surfaces, the two new cases being defined by a pair of “singular” Y -catenoids
been almost exclusively applied in the smooth setting: thus its adaptation to a class containing singular surfaces is notable. The only other application of the moving planes method in a non-smooth setting that we are aware of is the recent work [7,16]. However, in that work a posteriori regularity is derived from the moving planes method despite allowing a priori singularities. This is unlike our applications in which genuinely singular surfaces are symmetric examples; see Fig. 1. This introduction is organized as follows: in Sect. 1.2 we recall the rigidity theorems from [34]. In Sects. 1.3 and 1.4 we define Plateau surfaces and introduce a notion of orientability for them, that we call the cell structure condition. In Sect. 1.5 we state our main results, which extend Schoen’s rigidity theorems to minimal Plateau surfaces. Finally, in Sects. 1.6 and 1.7
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