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.



Area of metric spheres in Riemannian manifolds
The integral might be infinite.
Indeed, the function $A(r)$ can grow as fast as you want, but in this case the Ricci curvature in some of the radial direction have to go to $-\infty$ as $r\to\infty$. (Needless to say $A$ can grow faster than any polynomial.)


If Ricci curvature is bounded below then $A(r)$ has at most exponential growth --- this follows from the Bishop--Gromov inequality. In this case you integral has to be finite. 


