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Doubly Periodic Minimal Surface

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DoublyPeriodicMinimalSurface

A doubly periodic minimal surface is a periodic minimal surface invariant under two linearly independent translations but not under three linearly independent translations. For independent period vectors v_1 and v_2, the lattice

Lambda={mv_1+nv_2:m,n in Z}

acts by translations. The quotient is a minimal surface in R^3/Lambda, a flat torus times a line (Meeks and Pérez 2012, pp. 19-20).

The first of Scherk's minimal surfaces, illustrated above, satisfies

e^zcosy=cosx.

Its full period lattice is generated by (pi,pi,0) and (pi,-pi,0). Its quotient by this lattice has genus 0 and four ends (Meeks and Pérez 2012, pp. 25-26).

See also

Minimal Surface, Periodic Minimal Surface, Scherk's Minimal Surfaces, Singly Periodic Minimal Surface, Triply Periodic Minimal Surface

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References

Meeks, W. H., III and Pérez, J. A Survey on Classical Minimal Surface Theory. Providence, RI: Amer. Math. Soc., 2012. https://doi.org/10.1090/ulect/060.

Cite this as:

Weisstein, Eric W. "Doubly Periodic Minimal Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DoublyPeriodicMinimalSurface.html

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