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Symmetrization of starlike domains in Riemannian manifolds and a qualitative generalization of Bishop’s volume comparison theorem. (English) Zbl 1079.53025

Let \(M\) be a complete Riemannian manifold, let \(p \in M\) and let \(D\) be an open relatively compact domain in \(T_pM\) which is star-shaped with respect to the origin and such that the restriction of the exponential map \(\exp_p\) to \(D\) is a diffeomorphism onto its image. The Riemannian metric on \(M\) then lifts to a Riemannian metric \(g\) on \(D\). The \(g\)-volume of \(D\) can be calculated using a Riccati-equation along the radial geodesics through \(0\). The main goal of the paper is to investigate how this volume changes under specific changes of the metric on distance spheres.
The first such change replaces the metric on the distance spheres by a metric which is conformally equivalent to the standard metric while preserving the radial Ricci curvature. As in the classical Bishop volume comparison theorem, the volume form of the new metric is not smaller then the volume form of the original metric, with equality only if the two metrics coincide.
The second change replaces a metric whose restriction to distance spheres is conformally equivalent to the standard metric by a metric whose restriction to a distance sphere is a multiple of the standard metric while preserving the averages of the Ricci curvature over the distance spheres. In this case the volume form of the new metric is not bigger then the volume form of the original metric.

MSC:

53B20 Local Riemannian geometry
53C20 Global Riemannian geometry, including pinching
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