The stronger Arnold-Thom conjecture for unit tangents

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Let ff be an analytic function and let x(t)x(t) be a gradient flow line of ff. Suppose that x(t)x(t) has a limit point x∞x_{\infty} as t→∞t\to\infty. Stronger Arnold-Thom conjecture. The limit of the unit tangents

lim⁡t→∞x′(t)∣x′(t)∣\lim_{t\to\infty}\frac{x'(t)}{|x'(t)|}

exists. This strengthens the solved Thom gradient conjecture from convergence of secants to convergence of unit tangents. The source states that this stronger conjecture remains open.

References

Primary source

Tobias Holck Colding and William P. Minicozzi, “Analytical properties for degenerate equations”, arXiv:1804.08999 (2018).

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