Narrow Results

Following the publication online, it has been brought to the attention of the Editorial Board of Communications in Contemporary Mathematics (CCM) that the result of this paper was known since 1992 and appeared in the article by Luis Alías, Angel Ferrández and Pascual Lucas, Surfaces in the 3-dimensional Lorentz–Minkowski space satisfying $\Delta x=\mathrm{A}x+B$, Pacific J. Math.156(2) (1992) 201–208.

The information has been verified, and therefore the Editors-in-Chief of CCM have decided to retract the acceptance of this paper.

In this paper, we show that there is no timelike translation surface with non-minimal constant mean curvature $H$, constructed by the sum of a timelike plane curve and a timelike space curve in Minkowski 3-space $E_1^3$.

We discuss the problem of determining the de Rham, Dolbeault and algebraic cohomological dimension of ${{ℳ}}_g$, focusing on possible strategies of attack and then concentrating on exhaustion functions. In the final section, we explain how these techniques can be employed to provide a nontrivial upper bound for the Dolbeault cohomological dimension of ${{ℳ}}_g$.

In this paper, we classify translation surfaces in the three-dimensional simply isotropic space ${𝕀}_3^1$ under the condition ${\mathrm{\Delta }}^i{x}_i={\lambda }_i{x}_i$ where $\mathrm{\Delta }$ is the Laplace operator with respect to the first and second fundamental forms and $\lambda$ is a real number. We also give explicit forms of these surfaces.

In [M. K. Karacan, D. W. Yoon and B. Bukcu, Translation surfaces in the three-dimensional simply isotropic space ${𝕀}_3^1$, Int. J. Geom. Methods Mod. Phys.13(7) (2016) 1650088], there is a mistake in Theorem 5 that appeared in the paper. We here provide a correct theorem.

In this paper we will classify those translation surfaces in 𝔼³ involving polynomials which are Weingarten surfaces.