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Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies $$\mathrm{vol}(B_R(p)) \leq v(R)$$ for some function $v$. Can we then conclude some bounds on the volume growth of spheres, i.e. $$ \mathrm{vol}(\partial B_R(p)) \leq \tilde{v}(R)$$ for some other function $\tilde{v}$?

If this fails, under what additional geometric conditions can one obtain such a result?

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  • $\begingroup$ I believe it is true to say that when the Gaussian curvature is $\ge 0$, and hence there are no conjugate points, one has $$\frac{d}{dR} (\text{vol}(B_R(p))) = \text{vol}(\partial B_R(p))$$ $\endgroup$
    – Lee Mosher
    Commented Jun 9, 2017 at 20:54
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    $\begingroup$ this is true if there is a lower bound on Ricci curvature .This goes under the name of Bishop-Gromov comparison theorem .See for example H Karcher's paper Riemannian Comparison Constructions . $\endgroup$
    – Mohan Ramachandran
    Commented Jun 9, 2017 at 21:09

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