Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σ j (Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σ j (Σ, g) is the j-th nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ.

$d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

Modeling Symmetric Minimal Surfaces by Mesh Subdivision

Thanks to the great diffusion of additive manufacturing technologies, the interest in lattice structures is growing. Among them, minimal surfaces are characterized by zero mean curvature, allowing enhanced properties such as mechanical response and fluidynamic behavior. Recent works showed a method for geometric modeling triply periodic minimal surfaces (TPMS) based on subdivision surface. In this paper, the deviation between the subdivided TPMS and the implicit defined ones is investigated together with mechanical properties computed by numerical methods. As a result, a model of mechanical properties as a function of the TPMS thickness and relative density is proposed.