Minkowski minimal surfaces in ${\mathbb R}^3_1$ with minimal focal surfaces

Friedrich Manhart

Abstract: In Euclidean geometry a regular point on a focal surface of a minimal surface has non negative Gauss-curvature. So a focal surface of a minimal surface can never be a minimal surface. We classify the minimal surfaces in Minkowski 3-space the focal surfaces of which are minimal surfaces again. A (Euclidean) minimal surface $\Phi \subset \mathbb R^3$ only carries points with $K_e\leq 0$, where $K_e$ denotes the Gauss-curvature. Parametrizing $\Phi$ by a $C^2$-immersion $f:U \subseteq {\mathbb R}^2 \rightarrow f(U)=\Phi$ and assuming $K_e<0$, the principal curvatures (eigenvalues of the shape operator) are $k_{1,2}=\varphi(-K_e)^{1/2},(\varphi=\pm 1)$. Then the focal surfaces $\Psi_{\varphi}$ are parametrized by $z_{\varphi}=f+ (k_{1,2})^{-1}n_e=f+\varphi(-K_e)^{-1/2}n_e$, where $n_e$ is the unit normal vector. If a focal point is a regular point on $\Psi_{\varphi}$, then the Gauss-curvature of $\Psi_{\varphi}$ is positive. More precisely it is $K_e(\Psi_{\varphi})=-1/4\,K_e>0$ ([MAN]), so a focal surface of a minimal surface cannot be a minimal surface again. We will prove that in Minkowski (or Lorentz) 3-space ${\mathbb R}^3_1$ there are (up to scaling and Minkowski isometries) exactly two one-parameter families of minimal surfaces with this property.

Keywords: Minkowski minimal surfaces, focal surfaces, associated surfaces

Classification (MSC2000): 53A15; 53B30

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