Area of metric spheres in Riemannian manifolds 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.
 A: 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. 
A: I personally do not think that you can compute that integral explicitly in general just because the geodesic spheres could become quite ugly. $\newcommand{\bx}{\boldsymbol{x}}$ However, when $M$ has finite volume,  you can obtain   pretty good asymptotic estimates for the integral
$$ I(\lambda,\bx_0)=\int_M  e^{-\lambda d(\bx_0,\bx)^2}  dV_g(x), $$
as $\lambda\to \infty$. To obtain these  asymptotic estimates  you write  $dV_g(\bx)$ in normal coordinates $(t^1,\dotsc, t^n)$  at $\bx_0$ in the form
$$ dV_g(x) = \rho(t) dt^1\wedge\cdots \wedge dt^n. $$
The  Taylor  expansion of $\rho(t)$ at $t=0$  can be described explicitly in terms  of geometric invariants at $\bx_0$.    The  degree 4 Taylor polynomial of $\rho$ at $t=0$ is described in   Corollary 9.9 of A. Gray's book Tubes, 2nd Edition, Birhauser, 2006.   The Taylor  polynomials of degree $>4$ are  very complicated.
The integral you  are interested  could be computed on  spaces  with rich symmetries, but even in those cases it is not easy.
A: You could integrate the co-area formula by parts, to get:
$$\int_0^\infty 2re^{-r^2}V(x_0,r)dr$$
where $V(x_0,r)$ is the volume of the ball at $x_0$ of radius $r$, and then use comparison results for $V$.
