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Filling radius and short closed geodesics of the \(2\)-sphere. (English) Zbl 1064.53020
Let \((M,g)\) be a closed connected Riemannian manifold, and let \(d_g\) be the distance on \(M\). The map \(i: (M, d_g)\to (L^{\infty}(M), \|\cdot\|)\) defined by \(i(x)(\cdot)=d_g(x, \cdot)\) is an isometric embedding. Put \(U_{\delta}(M)\) to be the \(\delta\)-tubular neighborhood of \(i(M)\) in \(L^{\infty}(M)\). The filling radius is \(\text{FillRad}(M)=\inf \{\delta >0| (i_{\delta})_*([M])=0\in H_n(U_{\delta}(M))\}\).
The author shows that if \(M\) is a Riemannian \(2\)-sphere and scg\((M)\) is the length of the shortest closed geodesic on \(M\), then \(\text{FillRad}(M)\geq \frac{1}{12}\text{scg}(M)\). He also shows that \(\text{FillRad}(M)\geq \frac{1}{20}\overline L(M)\), where \(\overline L(M)\) is the length of the shortest nontrivial curve among the simple closed geodesics of index zero or one and the figure-eight geodesics of null index. As a corollary the author gets that Area\((M)\geq \frac{1}{20^2}\overline L(M)^2\). He also proves that Area\((M)\geq \frac{1}{C_1^2} \text{scg} (M)^2\) and that \(\operatorname{Diam}(M)\geq \frac{1}{C_2} \text{scg}(M),\) with \(C_1=12\) and \(C_2=4\).
This improves previous results of C. Croke who proved these inequalities with \(C_1=31\) and \(C_2=9\) [see “Area and the length of the shortest closed geodesic”, J. Differ. Geom. 27, No. 1, 1–21 (1988; Zbl 0642.53045)]. After having written the final version of the paper, the author learned that A. Nabutovsky and R. Rotman have independently proved the author’s inequality with \(C_2=4\) and proved a sharper estimate with \(C_1=8\) for the diameter of \(M\), see their work [“The length of the shortest closed geodesic on a 2-dimensional sphere”, Int. Math. Res. Not. 2002, No. 23, 1211–1222 (2002; Zbl 1003.53030)].

MSC:
53C20 Global Riemannian geometry, including pinching
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