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