## Explicit Björling surfaces with prescribed geometry.(English)Zbl 1403.53012

For an analytic curve $$c(t)$$ in $${\mathbb R^3}$$ and a unit vector field $$n(t)$$ along $$c(t)$$, orthogonal to $$c'(t)$$, the Björling formula $$\phi(z)=c(z)+\sqrt{-1}\,\int_0^z c'(t)\times n(t)dt$$ is a null curve in $$\mathbb{C}^3$$, i. e. the tangent vector of $$\phi(z)$$ has complex length 0. The real part $$f(u,v)$$ of $$\phi(u+\sqrt{-1}v)$$ defines a minimal surface in conformal parametrization containing $$c(t)$$, the core curve. Moreover, along $$c(t)$$, the surface normal agrees with $$n(t)$$.
If, for example, the core curve is the unit circle $$c(t)=(\cos(t),\sin(t),0)$$ and $$n(t)=c(t)$$, then the associated surface is the catenoid and if $$n(t)=\cos(a\,t)c(t)+(0,0,\sin(a\,t)), a\in{\mathbb Q}$$ rotates around $$c(t)$$ in a certain way, the corresponding surfaces are coiling around the circle. Especially, for $$a=1/2$$, a “minimal Möbius strip” and for integers $$a$$ certain circular helicoids are obtained.
Since an integration is involved, explicit results are obtained in rare cases. To find suitable settings for $$c(t)$$ and $$n(t)$$ the authors consider the class of polyexp functions, i. e., functions combined from terms of the form $$t^ne^{k\,t}$$, where $$n$$ is an integer and $$k\in\mathbb{C}$$. This class is preserved under algebraic operations, except division, as well as integration and differentiation. Therefore, if $$c(t)$$ and $$n(t)$$ are polyexp, the Björling formula gives explicit expressions of parametrized minimal surfaces, defined on the whole complex plane. To handle the orthogonality condition the following construction is applied: Let $$A$$ be a $$3\times3$$-matrix of polyexp functions with columns $$e_i(t)$$, $$i=1,2,3$$, such that $$A\,A^t$$ is a multiple $$\mu(t)^2\,E$$ of the identity matrix. Then, for linear functions $$\alpha(t)=a\,t+b$$ the Björling data $$c'(t)=e_1(t)$$ and $n(t)=\frac{1}{\mu(t)}\left(\cos(a\,t+b)\,e_2(t)+\sin(a\,t+b)\,e_3(t)\right),$ can be explicitely integrated and the corresponding parametric surface is defined and regular on the whole plane $$\mathbb{R}^2$$.
There are several ways to construct such “almost orthogonal” polyexp matrices $$A$$. One of them is the following: Consider a polyexp curve $${\mathbf q}=q_1(t)\,{\mathbf 1}+q_2(t)\,{\mathbf i}+q_3(t)\,{\mathbf j}+q_4(t)\,{\mathbf k}$$ in the algebra of quaternions. The linear action of $${\mathbf q}$$ on the subspace of imaginary quaternions defined by $${\mathbf v}\to{\mathbf q}{\mathbf v}\bar{{\mathbf q}}$$ gives a matrix $$A=A(q_1(t),q_2(t),q_3(t),q_4(t))$$ of this kind with $$\mu(t)=|{\mathbf q}(t)|$$. For $$q_1=\cos(t/2),q_2=q_3=0,q_4=-\sin(t/2)$$ the core curve is the unit circle in the $$xy$$-plane and various choices of $$\alpha(t)=a\,t+b$$ give circular helicoids. Core curves resulting from special choices of $${\mathbf q}(t)$$ are torus knots. Periodic surfaces are obtained from $${\mathbf q}_1=\cos(t/2){\mathbf j}+\sin(t/2){\mathbf k}$$, $${\mathbf q}_2=-\cos(t/2){\mathbf 1}+\sin(t/2){\mathbf k}$$ and $${\mathbf q}={\mathbf q}_1\cdot{\mathbf q}_2$$.
A second class of polyexp matrices $$A(v(t))=2v\cdot v^t-v^tv I_n$$ is obtained from $$180^\circ$$ rotations around a given polyexp vector function $$v(t)$$. Here, the choice of $$v=(x'(t),y'(t),\lambda)^t$$ with certain constants $$\lambda$$ and polyexp functions $$x(t),y(t)$$ produces core curves that are vertical lifts to $${\mathbb R}^3$$ of the plane curve $$(x(t),y(t),0)^t$$.
The Weierstraß data of these surface classes show that they are regular on $${\mathbb C}^\ast$$. A number of interesting surfaces are shown.

### MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q05 Minimal surfaces and optimization
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### References:

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