Regular homotopy and total curvature. I: Circle immersions into surfaces (Q2492037)

Let~\(\Sigma\) be a complete orientable surface with a Riemannian metric, and let~\(c\colon S^1\to\Sigma\) be an immersion. Two such immersions are called regularly homotopic if they are homotopic through a family of immersions. By the \(h\)-principle of Hirsch and Smale, this is the case if their unit tangent vectors define homotopic maps to the unit tangent bundle~\(U\Sigma\). The total absolute geodesic curvature~\(\kappa(c)\) is the integral of the absolute value of the geodesic curvature of~\(c\). The author investigates the infimum of~\(\kappa(c)\) in a given regular homotopy class. He proves that if the Gauß\ curvature~\(K\) is nonvanishing then the infimum can only be attained by geodesics, whereas if~\(K=0\), then the infimum is attained either by geodesics or by locally convex curves. In both cases, each local minimum of~\(\kappa\) is global in the given regular homotopy class. For general metrics, no such statement is possible. For flat~\(\Sigma\) the author also computes the homotopy type of the space of minimizers. Given two immersions~\(c_0\), \(c_1\colon S^1\to\Sigma\), the author considers~\(\max_t\kappa(c_t)\) for all regular homotopies~\(c_t\) connecting~\(c_0\) and~\(c_1\). For example he proves that on the round sphere, one can always achieve~\(\max_t\kappa(c_t) \leq\max\{\kappa(c_0),\kappa(c_1),2\pi+\epsilon\}\) if~\(\epsilon>0\). On the other hand, if~\(c_0\) runs~\(m\) times around a geodesic and \(c_1\) runs \(m+2\) times around a geodesic, then~\(\max_t\kappa(c_t)>2\pi\). In the proof, the author uses piecewise geodesics with curvature concentrations (PGC curves). These are closed piecewise geodesic curves, where the unit tangent vector of the incoming edge at each vertex~\(p\) is joined to the unit vector of the outgoing edge through a curve in~\(U_p\Sigma\). Each PGC curve~\(c\) defines a loop in~\(U\Sigma\), and~\(\kappa(c)\) is well-defined. In some cases, the infimum of the total absolute geodesic curvature in a regular homotopy class is attained by a PGC curve. [For Part II see Zbl 1114.53049.]