Examples of entire zero-mean curvature graphs of mixed-type in Lorentz-Minkowski space via Konderak's formulas

TL;DR

The paper investigates entire $ZMC$ graphs in Lorentz-Minkowski 3-space and uses Konderak's para-holomorphic representation formulas to generate explicit mixed-type examples. By employing Scherk-type input data, it constructs eight $ZMC$-surfaces and identifies four non-congruent connected instances, including mixed-type graphs, thereby proving the existence of a mixed-type entire $ZMC$-graph that is not a Kobayashi surface. It also demonstrates a mixed-type example over a light-like plane, suggesting a richer landscape of mixed-type $ZMC$-graphs beyond Kobayashi classifications. The work highlights global extendability considerations arising from non-isolated poles in the para-holomorphic framework and broadens the catalog of known entire $ZMC$-graphs in $R^3_1$.

Abstract

Using Konderak's representation formula, we construct an entire zero-mean curvature graph of mixed-type in Lorentz-Minkowski 3-space over a space-like plane, which does not belong to the class of "Kobayashi surfaces". We also point out the existence of an entire zero-mean curvature graph of mixed-type in Lorentz-Minkowski space over a light-like plane. These examples suggest that entire mixed-type zero-mean curvature graphs contain an unexpectedly large number of interesting examples.

Figures (2)

• Figure 1: The Enneper-type surface ${\mathcal{E}}_4$ (left) and the Scherk-type surfaces ${\mathcal{S}}_1$ (center) and ${\mathcal{S}}'_1$ (right)
• Figure 2: The Scherk-type surfaces $\mathcal{S}_2$ (left) and $\mathcal{S}'_2$ (left)

Theorems & Definitions (20)

• Theorem 1.1
• Proposition 1.2
• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma A.1
• proof
• Proposition A.2