[Paper Review] Behavior of the Gaussian curvature of timelike minimal surfaces with singularities

Shintaro Akamine|arXiv (Cornell University)|Jan 1, 2017

Geometric Analysis and Curvature Flows2 references7 citations

TL;DR

This paper investigates the sign and behavior of Gaussian curvature on timelike minimal surfaces in Lorentz-Minkowski space, showing that the curvature's sign is determined by the degeneracy and relative orientations of the two null curves generating the surface. It establishes that near non-degenerate singular points—such as cuspidal edges, swallowtails, or cuspidal cross caps—the Gaussian curvature diverges to $-\infty$ or $+\infty$, depending on whether the surface is a front or not, with precise criteria derived from the geometry of the generating curves and their pseudo-arclength parameters.

ABSTRACT

In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points.

Motivation & Objective

• To determine the sign of Gaussian curvature on timelike minimal surfaces in $\mathbb{L}^3$ using the geometric properties of their generating null curves.
• To characterize flat points (umbilic and quasi-umbilic) via the non-degeneracy of the generating null curves.
• To establish criteria for the sign of Gaussian curvature near non-degenerate singular points such as cuspidal edges, swallowtails, and cuspidal cross caps.
• To analyze curvature behavior near singularities, particularly the divergence to $-\infty$ or $+\infty$, depending on whether the surface is a front or not.

Proposed method

• Uses the representation of timelike minimal surfaces as the sum of two null curves, leveraging the Weierstrass-type representation in $\mathbb{L}^3$.
• Applies the notion of pseudo-arclength parameters for null curves to construct conformal curvature line and asymptotic coordinate systems.
• Employs differential geometric criteria from frontal and front theory, particularly Fact A.10 and Fact A.11, to classify singular points.
• Relies on the induced metric from $\mathbb{L}^3$ and compares curvature behavior with the Euclidean case, noting the sign reversal between $\mathbb{E}^3$ and $\mathbb{L}^3$ Gaussian curvatures.
• Uses the conjugate minface duality to relate singular point types between a minface and its conjugate, as formalized in Fact A.12.
• Derives explicit expressions for the determinant of the Jacobian of the singular curve and null direction field to classify singularities.

Experimental results

Research questions

• RQ1 How is the sign of the Gaussian curvature of a timelike minimal surface in $\mathbb{L}^3$ determined by the degeneracy and orientations of its generating null curves?
• RQ2 What is the behavior of the Gaussian curvature near non-degenerate singular points such as cuspidal edges, swallowtails, and cuspidal cross caps?
• RQ3 How does the front condition at a singular point affect the sign and divergence of the Gaussian curvature?
• RQ4 What is the relationship between the singular curvature on a cuspidal edge and the Gaussian curvature with respect to the $\mathbb{L}^3$ metric?

Key findings

• The sign of the Gaussian curvature of a timelike minimal surface is determined solely by the degeneracy and relative orientations of its two generating null curves.
• At a non-degenerate singular point that is a cuspidal edge, there are no umbilic or quasi-umbilic points nearby, and the Gaussian curvature diverges to $-\infty$.
• If the surface is a front at a non-cuspidal non-degenerate singular point, the Gaussian curvature is negative nearby and diverges to $-\infty$ as the point is approached.
• If the surface is not a front at a non-degenerate singular point, the Gaussian curvature is positive nearby and diverges to $+\infty$ as the point is approached.
• The Gaussian curvature with respect to the $\mathbb{L}^3$ metric near a cuspidal edge has the same sign as the singular curvature, a result stronger than the one in [20] due to the sign reversal between $\mathbb{E}^3$ and $\mathbb{L}^3$ metrics.
• The conjugate minface duality swaps the types of singular points: a cuspidal edge maps to a cuspidal edge, while a swallowtail maps to a cuspidal cross cap and vice versa.