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Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,d)$ is Ahlfors $n$-regular, meaning that: there exist constants $L,U>0$ for which $$ L r^n\leq \mu(B(x,r))\leq U r^n$$ where $\mu$ is the $n$-dimensional Hausdorff measure and $B(x,r)$ denotes the ball in $(X,d)$ about $x\in X$ of radius $r>0$ (i.e. each $\mu(B(x,r))\in \Theta( r^n)$).

  1. Where can I find a proof or reference for this fact?
  2. Can the constants $L,U$ be stated using the curvature and dimension of $(M,g)$?
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    $\begingroup$ Non-negative Ricci curvature implies the weaker doubling condition (see fabricebaudoin.wordpress.com/2013/10/03/… ) but not Ahlfors regularity. For example, an infinite cylinder is not Ahlfors regular. $\endgroup$
    – user142382
    Commented Aug 19, 2021 at 17:44
  • $\begingroup$ Ah, so are there examples of Ahlfors-Regular Riemannian manifolds (besides Euclidean space)? $\endgroup$
    – ABIM
    Commented Aug 21, 2021 at 12:09
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    $\begingroup$ There is no such issue when $M$ is compact (in the definition that I have seen of Ahlfors regularity, r is taken to be less than the diameter), and indeed every compact Riemannian manifold will be Ahlfors regular. $\endgroup$
    – Nicolast
    Commented Aug 25, 2021 at 13:29
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    $\begingroup$ For $0\le r\le D=diam(M)$, you can just take $L$ and $U$ as the min and max of the continuous function $$f(x,r)=\begin{cases}\frac1r \mu(B(x,r))^{1/n}\ \text{ if }\ r>0\\ \frac\pi2 \Gamma(\frac n2+1)^{-1/n}\ \text{ if }\ r=0\end{cases}$$ over the compact region $M\times[0,D]$, where the second case corresponds to the Euclidean volume. $\endgroup$
    – user44143
    Commented Sep 5, 2021 at 15:23
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    $\begingroup$ @BLBA. The reason to distinguish the case $r=0$ is that the formula for $r>0$ is not defined for that case. Also it is a theorem that a continuous function with positive values at every point of a compact region has a positive minimum; you just have to prove the continuity by writing $\mu(B(x,r))$ as an integral over the tangent vectors at $x$ which correspond to geodesics between $x$ and points of the ball. $\endgroup$
    – user44143
    Commented Sep 8, 2021 at 16:06

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