Instability of a Möbius strip minimal surface and a link with systolic geometry.

Adriana I Pesci,Raymond E Goldstein,H Keith Moffatt,Gareth P Alexander

doi:10.1103/physrevlett.114.127801

Adriana I Pesci, Raymond E Goldstein + Show 2 more

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https://doi.org/10.1103/physrevlett.114.127801

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Abstract

We describe the first analytically tractable example of an instability of a nonorientable minimal surface under parametric variation of its boundary. A one-parameter family of incomplete Meeks Möbius surfaces is defined and shown to exhibit an instability threshold as the bounding curve is opened up from a double-covering of the circle. Numerical and analytical methods are used to determine the instability threshold by solution of the Jacobi equation on the double covering of the surface. The unstable eigenmode shows excellent qualitative agreement with that found experimentally for a closely related surface. A connection is proposed between systolic geometry and the instability by showing that the shortest noncontractable closed geodesic on the surface (the systolic curve) passes near the maximum of the unstable eigenmode.

Highlights

The subject of this paper lies at the intersection of several branches of physics and mathematics: minimal surfaces, soap films, topological transitions, computation, and shortest closed noncontractible geodesics on surfaces, known as systolic curves [1]
Euler’s discovery in 1744 [2] of the catenoid as the area-minimizing surface spanning two circular loops effectively began the study of minimal surfaces, which are known to play a role in areas as diverse as soap films [3], supramolecular assemblies [4], defect structures [5], as well as general relativity [6], string theory [7], and even architecture [8]
In a further connection between systolic geometry and stability theory, we show that the systolic curve on the unstable minimal surface approaches the maximum of the unstable eigenfunction, confirming one’s intuition that the instability begins at the narrowest part of the neck

Summary

Introduction

A one-parameter family of incomplete Meeks Mobius surfaces is defined and shown to exhibit an instability threshold as the bounding curve is opened up from a double-covering of the circle. Numerical and analytical methods are used to determine the instability threshold by solution of the Jacobi equation on the double-cover of the surface.

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