I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I can think of is to use some coarea theorem, leading to $\int _0 ^\infty \mathbb{e} ^{-r^2} A(x_0, r) \mathbb{d}r$, where $A(x_0, r)$ is the area of the metric sphere of center $x_0$ and radius $r$. The issue now is to have some estimate of $A$. Surprisingly, even though there are plenty of comparison theorems for the volume of metric balls, I haven't been able to find anything usable on the area of spheres. Therefore, the questions:

- how would you approach this integral (if not by the coarea formula)?
- is it finite? (I assume so)
- is $A$ a polynomial in $r$ (of degree $n-1$)? If so, I would expect it not to have a free term, but what about its leading coefficient?
- can $A$ be computed explicitly in space forms (other than Euclidian, of course)?
- are there comparison theorems for $A$?

Thank you.