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Regular homotopy and total curvature. I: Circle immersions into surfaces. (English) Zbl 1113.53041

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.]

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A04 Curves in Euclidean and related spaces
57R42 Immersions in differential topology

Citations:

Zbl 1114.53049
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References:

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