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**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.

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.

Reviewer: Hubert Gollek (Berlin)

### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

49Q05 | Minimal surfaces and optimization |

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\textit{R. López} and \textit{M. Weber}, Mich. Math. J. 67, No. 3, 561--584 (2018; Zbl 1403.53012)

### References:

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