[Paper Review] Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts

Luciana F. Martins, Kentaro Saji+2 more|arXiv (Cornell University)|Aug 9, 2013

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper introduces intrinsic geometric invariants—cuspidal curvature $\kappa_c$ and limiting normal curvature $\kappa_\nu$—for non-degenerate singularities on wave fronts in Riemannian 3-manifolds. It establishes that the product $\kappa_\Pi = \kappa_\nu \kappa_c$ is an intrinsic invariant determining the boundedness of Gaussian curvature, with $\kappa_\Pi = 0$ if and only if the Gaussian curvature is rationally bounded near the singularity.

ABSTRACT

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $κ_Π$ called the product curvature (resp. $μ_Π$ called normalized product curvature) of $κ_c$ (resp. $μ_c$) and the limiting normal curvature $κ_ν$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $κ_Π$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.

Motivation & Objective

• To define intrinsic geometric invariants for non-degenerate singular points on wave fronts in Riemannian 3-manifolds.
• To clarify the relationship between the behavior of Gaussian and mean curvature near cuspidal edges and swallowtails and intrinsic differential invariants.
• To establish that the product curvature $\kappa_\Pi = \kappa_\nu \kappa_c$ is an intrinsic invariant determining the boundedness of Gaussian curvature.
• To generalize the concept of limiting normal curvature $\kappa_\nu$ to arbitrary rank-one singular points on wave fronts.
• To analyze the limiting behavior of $\kappa_c$ and $\kappa_\nu$ when cuspidal edges accumulate toward other singularities.

Proposed method

• Defining the cuspidal curvature $\kappa_c$ along cuspidal edges as the first coefficient of the divergent term in the mean curvature function expansion.
• Introducing the limiting normal curvature $\kappa_\nu$ as a geometric invariant derived from the first fundamental form and the Gauss map.
• Constructing the product curvature $\kappa_\Pi = \kappa_\nu \kappa_c$ and proving it is an intrinsic invariant via coordinate-invariant expressions in terms of the first fundamental form.
• Using blow-up techniques to define rational boundedness and continuity of curvature functions near singularities.
• Applying the Weingarten formula and Gauss equation to relate the signed area element $d\hat{A}$ to the Gaussian curvature $K$ and the Gauss map.
• Extending the analysis to hyperbolic 3-space $H^3$ by considering the Gauss map into de Sitter space $S^3_1$ and proving analogous results for $\kappa_\nu$.

Experimental results

Research questions

• RQ1 What intrinsic invariants govern the behavior of Gaussian and mean curvature near non-degenerate singularities on wave fronts?
• RQ2 How is the product curvature $\kappa_\Pi = \kappa_\nu \kappa_c$ related to the boundedness of Gaussian curvature?
• RQ3 Under what conditions is the Gaussian curvature rationally bounded near a cuspidal edge or swallowtail singularity?
• RQ4 How does the limiting normal curvature $\kappa_\nu$ relate to the singularity of the Gauss map?
• RQ5 What happens to the curvatures $\kappa_c$ and $\kappa_\nu$ when cuspidal edges accumulate toward other singularities?

Key findings

• The product curvature $\kappa_\Pi = \kappa_\nu \kappa_c$ is an intrinsic invariant of the surface, expressible solely in terms of the first fundamental form.
• The Gaussian curvature $K$ is rationally bounded at a non-degenerate singular point $p$ if and only if the limiting normal curvature $\kappa_\nu(p) = 0$.
• For a front in $\mathbb{R}^3$, the extension of the 2-form $K\,d\hat{A}$ vanishes at $p$ if and only if $\kappa_\nu(p) = 0$, and this is equivalent to the Gauss map having a singularity at $p$.
• The cuspidal curvature $\kappa_c$ coincides with the curvature of the cusp in the plane section orthogonal to the tangential direction of the cuspidal edge.
• The normalized product curvature $\mu_\Pi = \kappa_\nu \mu_c$ at swallowtail singularities is also an intrinsic invariant and controls the boundedness of Gaussian curvature.
• When cuspidal edges accumulate toward other singularities, the limiting normal curvature $\kappa_\nu$ and cuspidal curvature $\kappa_c$ exhibit controlled blow-up behavior, with $\kappa_\Pi$ remaining bounded if and only if $\kappa_\nu \to 0$.