An obstruction to the existence of immersed curves of prescribed curvature (Q845810)

The main result is the following: Theorem 1.2. Let \(H:\mathbb{R}^2\to\mathbb{R}\) be a positive Lipschitz function such that there exists \(\overrightarrow{e}\in\mathbb{R}^2\) verifying \[ \langle\nabla H(x),\overrightarrow{e}\rangle>0,\quad \text{for almost every}\;\;x\in\mathbb{R}^2. \] Then there exists no \(C^1\)-closed curve immersed in \(\mathbb{R}^2\) whose curvature at every point \(x\) is \(H(x)\). The related result for \(C^1\)-closed hypersurfaces embedded in \(\mathbb{R}^n\) of mean curvature \(H\), namely the theorem 1.1 in this paper, is proved by the first author in his thesis (PIMS, UBC, Canada). The authors want to investigate a generalization of the result of the first author in the immersed case.