Extension of Krust theorem and deformations of minimal surfaces (Q2088029)

The Krust theorem asserts that if a minimal surface in Euclidean space \(\mathbb{R}^3\) is the graph of a function over a convex domain, then all its associated surfaces also are graphs. This result was recently extended to the Lorentzian space \(\mathbb{L}^3\) (see [\textit{R. López}, Bull. Malays. Math. Sci. Soc. (2) 44, No. 2, 683--692 (2021; Zbl 1466.53009)]). If the initial minimal surface \(X\) in \(\mathbb{R}^3\) has a Weierstrass data \((F,G)\), the associated minimal surface is determined by \((G,e^{i\theta}F)\). By introducing new parameters \(\lambda\in (0,\infty)\) and \(c\in \mathbb{R}\), the surfaces \[ X_{\theta,\lambda,c}=\Re\int^w(1-c\lambda^2 G^2,-i(1+c\lambda^2 G^2),2\lambda G)\frac{e^{i\theta}}{\lambda} F \, dw \] provide minimal surfaces in Euclidean space if \(c=1\) (in Lorentzian space \(c=-1\), isotropic space \(c=0\), respectively). The parameter \(\theta\) gives the deformation by the associated surfaces and the parameter \(\lambda\) corresponds with the López-Ros deformation. In this paper, it is proved that under certain natural restrictions on the parameters \(c\) and \(\lambda\), if a minimal surface \(X_{\theta_0,\lambda_0,c_0}\) is a graph on a convex domain, then all minimal surfaces \(X_{\theta,\lambda,c}\) also are graphs.