Balanced parametrizations of boundaries of three-dimensional convex cones. (arXiv:1810.06675v1 [math.DG])

Let $K \subset \mathbb R^3$ be a regular convex cone with positively curved boundary of class $C^k$, $k \geq 5$. The image of the boundary $\partial K$ in the real projective plane is a simple closed convex curve $\gamma$ of class $C^k$ without inflection points. Due to the presence of sextactic points $\gamma$ does not possess a global parametrization by projective arc length. In general it will not possess a global periodic Forsyth-Laguerre parametrization either, i.e., it is not the projective image of a periodic vector-valued solution $y(t)$ of the ordinary differential equation (ODE) $y''' + \beta \cdot y = 0$, where $\beta$ is a periodic function. We show that $\gamma$ possesses a periodic Forsyth-Laguerre type global parametrization of class $C^{k-1}$ as the projective image of a solution $y(t)$ of the ODE $y''' + 2\alpha \cdot y' + \beta \cdot y = 0$, where $\alpha \leq \frac12$ is a constant depending on the cone $K$ and $\beta$ is a $2\pi$-periodic function of class $C^{k-5}$ 查看全文>>