Small volume on big 𝑛-spheres

Croke, Christopher

doi:10.1090/S0002-9939-07-09079-X

Small volume on big $n$-spheres
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by Christopher B. Croke PDF

Proc. Amer. Math. Soc. 136 (2008), 715-717 Request permission

Abstract:

We consider Riemannian metrics on the $n$-sphere for $n\geq 3$ such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the $2$-dimensional case where Berger has shown that $Area\geq 1/2$.

References

Marcel Berger, Volume et rayon d’injectivité dans les variétés riemanniennes de dimension $3$, Osaka Math. J. 14 (1977), no. 1, 191–200 (French). MR 467595
Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287
Christopher B. Croke, Curvature free volume estimates, Invent. Math. 76 (1984), no. 3, 515–521. MR 746540, DOI 10.1007/BF01388471
Christopher B. Croke, The volume and lengths on a three sphere, Comm. Anal. Geom. 10 (2002), no. 3, 467–474. MR 1912255, DOI 10.4310/CAG.2002.v10.n3.a2
Christopher B. Croke and Mikhail Katz, Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 109–137. MR 2039987, DOI 10.4310/SDG.2003.v8.n1.a4
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
S. V. Ivanov, Gromov-Hausdorff convergence and volumes of manifolds, Algebra i Analiz 9 (1997), no. 5, 65–83 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 5, 945–959. MR 1604389
P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. MR 48886