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- W2023797238 abstract "We prove that $${mathcal{C}}^infty$$ curves of constant curvature satisfy, in the sense of Gromov, the relative $${mathcal{C}}^1$$ -dense h-principle in the space of immersed curves in Euclidean space R n ≥ 3. In particular, in the isotopy class of any given $${mathcal{C}}^1$$ knot f there exists a $${mathcal{C}}^infty$$ knot f͂ of constant curvature which is $${mathcal{C}}^1$$ -close to f. More importantly, we show that if f is $${mathcal{C}}^2$$ , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull." @default.
- W2023797238 created "2016-06-24" @default.
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- W2023797238 date "2007-07-24" @default.
- W2023797238 modified "2023-10-18" @default.
- W2023797238 title "h-Principles for curves and knots of constant curvature" @default.
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- W2023797238 doi "https://doi.org/10.1007/s10711-007-9151-y" @default.
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