Let $n$ - dimension $\geq 3$.

Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \epsilon_0$.

Consider the Green's function on $B_\epsilon(0)$ ( i.g. verifies that $\Delta G = \delta_y$ and $G=0$ on the boundary. G is also positive, smooth and well defined of the diagonal).

Is it possible to get the following upper bound

$$ G(x,y) \leq C(n) \rho(x,y)^{2-n}. $$

It is known that this estimate holds near the singularity (even for a general compact subdomain of $(M,g)$); see Schoen and Yau for instance.

Is it true for all $x\neq y$, not just near the singularity?

N.B. In $\mathbb{R}^n$, we know that $G(x,y) \leq \mbox{ Fundamental solution } \leq C r^{2-n}$; it is a consequence of the max principle. In short, I am trying to get such estimates for a geodesic ball on manifolds.

Do we have an explicit formula for the Green's function of a geodesic ball ? Can we derive such bound from it?

  • $\begingroup$ On a compact manifold, how is a bound other than near the singularity an issue? Am I missing something here? $\endgroup$ – Michael Renardy May 27 '13 at 15:39
  • $\begingroup$ I need a global estimate, not a local one. If I have an estimate near the singularity, does that mean that it holds on the whole manifold? $\endgroup$ – Henry May 27 '13 at 15:47
  • $\begingroup$ I must add that if I use compactness to extend, my constant C will depend on $\Omega$ and I do not want that. $\endgroup$ – Henry May 27 '13 at 15:50
  • $\begingroup$ For instance, if I say that $G(x,y) \leq C(n) \rho(x,y)^{2-n} + M Vol (\B_\epsilon\setminus \B_r (y) ),$ it will not be enough. I really need a bound $C(n) \rho(x,y)^{2-n}$ for $B_\epsilon$. $\endgroup$ – Henry May 27 '13 at 15:57
  • $\begingroup$ Maybe I can ask a related question. It is known in $\mathbb{R}^n$ that $G(x,y)$ is given by a regular part (harmonic function) and by the Fondamental solution. Is there something equivalent for a Green's function on a compact subdomain (or geodesic ball) of a given manifold $(M,g)$? $\endgroup$ – user34432 May 27 '13 at 16:52

I suggest you look at: http://www.jstor.org/stable/2374588

| cite | improve this answer | |
  • $\begingroup$ Yes. I know about that paper. It gives the bound for the singularity for instance. However, I did not see anything else that could give me an answer. Did I miss something? $\endgroup$ – user34432 May 27 '13 at 17:56
  • $\begingroup$ Define $M(r) = max_{\partial B_r(y)} G(r), r = \rho(x,y)$. It is known that : i) near its singularity, $M(r)\leq C(n) r^{2-n}$, e.g. for small $r$ ii) M(r) decreases as $r$ increases My question is : iii) is the following hold : $M(r) \leq C(n) r^{2-n}$ for all $r\leq R$ of my geodesic ball. Note that in $\mathbb{R}^n, G(x,y) \leq C(n) r^{2-n}$ for all $r\leq R$ in a ball of radius R for instance. This follows from the fact that the Green's function is "bounded" by the fondamental solution of the Laplace operator. $\endgroup$ – user34432 May 27 '13 at 18:08

I don't know if this helps you...

Consider the warped product metric on $M^n = R \times S^{n-1}$

$ds^2 = dr^2 + f(r)^2 ds_{S^{n-1}}.$

Assume $u=u(r)$, where $r$ is the distance function from some fixed point.


$\nabla u = u'(r) \nabla r$

$\Delta u = u''(r)\Delta r + u'(r)^2$

Sometimes you can find Green function solving the EDO

$\Delta u = 0.$


$\Delta r = (n-1) \frac{f'(r)}{f(r)}$,


$(n-1)u''(r) \frac{f'(r)}{f(r)} + u'(r)^2 =0$.

| cite | improve this answer | |
  • $\begingroup$ Indeed, I knew about that approach. However, I do not want to consider a specific metric. I was trying to find in the literature a formula for the green's function of a geodesic ball, which seems unknown. I was wondering if it were indeed the case and if some bounds were known instead. $\endgroup$ – Henry May 27 '13 at 16:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.