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By the classical Li–Yau inequality, an immersion of a closed surface inℝn{\mathbb{R}^{n}}with Willmore energy below8⁢π{8\pi}has to be embedded. We discuss analogous results for curves inℝ2{\mathbb{R}^{2}}, involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

A Li–Yau inequality for the 1-dimensional Willmore energy