In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other. Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.
Scherk's first surface Scherk's first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near z = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines.
Construction of a simple Scherk surface
Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number n, find a minimal surface Σn as the graph of some function
u n : ( − π 2 , + π 2 ) × ( − π 2 , + π 2 ) → R {\displaystyle u_{n}:\left(-{\frac {\pi }{2}},+{\frac {\pi }{2}}\right)\times \left(-{\frac {\pi }{2}},+{\frac {\pi }{2}}\right)\to \mathbb {R} }
such that
lim y → ± π / 2 u n ( x , y ) = + n for − π 2 < x < + π 2 , {\displaystyle \lim _{y\to \pm \pi /2}u_{n}\left(x,y\right)=+n{\text{ for }}-{\frac {\pi }{2}}<x<+{\frac {\pi }{2}},}
lim x → ± π / 2 u n ( x , y ) = − n for − π 2 < y < + π 2 . {\displaystyle \lim _{x\to \pm \pi /2}u_{n}\left(x,y\right)=-n{\text{ for }}-{\frac {\pi }{2}}<y<+{\frac {\pi }{2}}.}
That is, un satisfies the minimal surface equation
d i v ( ∇ u n ( x , y ) 1 + | ∇ u n ( x , y ) | 2 ) ≡ 0 {\displaystyle \mathrm {div} \left({\frac {\nabla u_{n}(x,y)}{\sqrt {1+|\nabla u_{n}(x,y)|^{2}}}}\right)\equiv 0}
and
Σ n = { ( x , y , u n ( x , y ) ) ∈ R 3 | − π 2 < x , y < + π 2 } . {\displaystyle \Sigma _{n}=\left\{(x,y,u_{n}(x,y))\in \mathbb {R} ^{3}\left|-{\frac {\pi }{2}}<x,y<+{\frac {\pi }{2}}\right.\right\}.}
What, if anything, is the limiting surface as n tends to infinity? The answer was given by H. Scherk in 1834: the limiting surface Σ is the graph of
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