Flat curves (Q1693386)

From the authors' introduction: ``It will be seen that an algebraic plane curve, viewed as a Riemann surface, may be assigned a polyhedral geometry in a natural way.'' Theorem 1.1. If \(\mathcal{C} \subset \mathbb{C}\mathbb P^{2}\) is an irreducible, real algebraic curve of degree \(d \leq 7\), then \((\mathcal{C}, dx^{2}+dy^{2})\) is flat and compact if and only if \(\mathcal{C}\) is the sextic trefoil. ``For one interpretation of the theorem, regard an arc length parametrization of a plane curve \(\mathcal{C} \subset \mathbb{R}^{2}\) as a Riemannian covering of one-dimensional manifolds \(\gamma : \mathbb{R}\rightarrow \mathcal{C}\).'' Corollary 1.2. Let \(\mathcal{C}\) as above, but with positive genus. Let \(x(s),y(s)\) parameterize an arc of \(\mathcal{C}\) by unit speed. If \(x(s)\) and \(y(s)\) extend meromorphically to all \(s \in \mathbb{C},\) then \(\mathcal{C}\) is the sextic trefoil, an immersed torus. The authors characterize the sextic trefoil among plane curves of low degree: first as a complex curve with compact, flat geometry; and then as a curve with meromorphic arc length parameterization.