Assume that $\rho$ and $\rho'$ are conformal metrics on the unit disk which is a geodesic disk of radius $1$ w.r.t. both metrics $\rho$ and $\rho'$, and assume that $\rho'$ has a constant Gauss curvature $K'$ which is greater than the curvature of $\rho$. Let $d$ and $d'$ be the corresponding distance functions defined on the unit disk. Which is the best reference for the inequality $d(0,a)\le d'(0,a)$.
2 Answers
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See Bob Osserman's 1999 Notices article (which is beautifully written, and copiously referenced).
answered Aug 6, 2015 at 19:57
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$\begingroup$ I have seen that paper, but i guess that the result have been obtained before? $\endgroup$– miskoAug 6, 2015 at 19:59
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$\begingroup$ @misko Osserman has very extensive references, and under various hypotheses, there are versions of the results due to Ahlfors, Yau, and Troyanov (the last of which not long before Osserman's paper). So, whatever Osserman attributes to other people is due to them, and whatever he does not is due to him :) $\endgroup$ Aug 6, 2015 at 20:02
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The statement is not correct. A bigger sphere has smaller curvature. Half of a sphere is a conformal disk.
answered Aug 7, 2015 at 2:51