In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing osculating frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly simplifies if the curve is a polynomial one. Further, polynomial PB curves of the degree $\ge 7$ and rational PB curves of the degree $\ge 6$ that possess rational rotation-minimizing osculating frames are derived, and it is shown that no lower degree curves, constructed from quadratic quaternion polynomials, with such a property exist.
COBISS.SI-ID: 1537835204
In this paper we show that for every conformal minimal immersion $u \colon M \to \mathbb{R}^3$ from an open Riemann surface $M$ to $\mathbb{R}^3$ there exists a smooth isotopy $u_t \colon M \to \mathbb{R}^3$ $(t \in [0,1])$ of conformal minimal immersions, with $u_0 = u$, such that $u_1$ is the real part of a holomorphic null curve $M \to \mathbb{C}^3$ (i.e. $u_1$ has vanishing flux). Furthermore, if $u$ is nonflat then $u_1$ can be chosen to have any prescribed flux and to be complete.
COBISS.SI-ID: 17540953
We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $C^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^n$, for $N$ large enough, thereby answering a question of Alarcón and Forstnerič, 2013.
COBISS.SI-ID: 17333081