On the volume of metric balls

Croke, Christopher B.

doi:10.1090/S0002-9939-1983-0702295-3

On the volume of metric balls
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by Christopher B. Croke PDF

Proc. Amer. Math. Soc. 88 (1983), 660-664 Request permission

Abstract:

In this paper we consider metrics of the form $d{s^2} = d{r^2} + {f^2}(r,\theta )d{\theta ^2}$ on a ball of dimension $n \geqslant 3$. We show that if the diameters (geodesics through the origin) minimize length then the volume of the ball is larger than the volume of the hemisphere of the corresponding round sphere. This relates to a conjecture first considered by Marcel Berger. We also give examples in all dimensions of radially symmetric metrics on balls of radius $\pi /2$ having arbitrarily small volume and yet having no pair of points conjugate along a diameter.

References

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