Berger-Kazdan comparison theorem

In mathematics, the Berger-Kazdan comparison theorem is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the "m"-dimensional sphere with its usual "round" metric. The theorem is named after the mathematicians Marcel Berger and Jerry Kazdan.

tatement of the theorem

Let ("M", "g") be a compact "m"-dimensional Riemannian manifold with injectivity radius inj("M"). Let vol denote the volume form on "M" and let "c"_"m"("r") denote the volume of the standard "m"-dimensional sphere of radius "r". Then

:$mathrm\{vol\}\; (M)\; geq\; frac\{c\_\{m\}\; (mathrm\{inj\}(M))\}\{pi\},$

with equality if and only if ("M", "g") is isometric to the "m"-sphere S^"m" with its usual round metric.

References

*cite book|last = Berger|first = Marcel|authorlink=Marcel Berger|coauthors = Kazdan, Jerry L.|chapter = A Sturm-Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds|title = Proceedings of Second International Conference on General Inequalities, 1978|publisher = Birkhauser|year = 1980|pages = 367–377
*cite journal|last = Kodani|first = Shigeru|title = An Estimate on the Volume of Metric Balls|journal = Kodai Mathematical Journal|volume = 11|issue = 2|date = 1988|pages = 300–305|url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kmj/1138038881|doi = 10.2996/kmj/1138038881

External links

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