## 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

### Keywords:

total geodesic curvature; circle immersions; PGC curve

Zbl 1114.53049
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### References:

 [1] V I Arnol’d, Plane curves, their invariants, perestroikas and classifications, Adv. Soviet Math. 21, Amer. Math. Soc. (1994) 33 · Zbl 0864.57027 [2] T Ekholm, Regular homotopy and total curvature II: sphere immersions into $$3$$-space, Algebr. Geom. Topol. 6 (2006) 493 · Zbl 1114.53049 [3] M Gromov, Partial differential relations, Ergebnisse series 9, Springer (1986) · Zbl 0651.53001 [4] M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 · Zbl 0113.17202 [5] A V Inshakov, Invariants of types $$j^+, j^-$$ and $$\mathrm st$$ of smooth curves on two-dimensional manifolds, Funktsional. Anal. i Prilozhen. 33 (1999) 35, 96 · Zbl 0957.57019 [6] J Jost, Riemannian geometry and geometric analysis, Universitext, Springer (1998) · Zbl 0997.53500 [7] L A Lyusternik, A I Fet, Variational problems on closed manifolds, Doklady Akad. Nauk SSSR $$($$N.S.$$)$$ 81 (1951) 17 [8] S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. $$(2)$$ 69 (1959) 327 · Zbl 0089.18201 [9] V Tchernov, Arnold-type invariants of wave fronts on surfaces, Topology 41 (2002) 1 · Zbl 1017.57006
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