Let $(M,g)$ be a closed Riemann surface, and $p\in M$ any point. Let $G_p$ be the Green's kernel of the Laplacian with singularity at $p$, normalized to have zero average. Then locally (in some normal coordinates centered at $p$) $$G_p(x) = \log|x| + A + O(|x|).$$ Is there some geometric significance of the constant $A$? For instance the Green's function of the conformal Laplacian (say in dimension 3) is related to the ADM mass of some asymptotically flat manifold, but I could not find any result for the usual Laplacian. Any reference would be appreciated!
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3$\begingroup$ A change of co-ordinates changes $A$. For example, if $x=\lambda x'$ for some constant $\lambda$, then $\log|x|=\log|x'|+\log|\lambda|$. $\endgroup$– macbethCommented Jun 8, 2017 at 18:34
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1$\begingroup$ Oh of course. Thanks! I should require the coordinates to be normal coordinates, so that at least takes care of scaling. $\endgroup$– Poincare-LelongCommented Jun 8, 2017 at 20:27
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4$\begingroup$ See examples of page 7, and 8 ms.u-tokyo.ac.jp/~hirachi/scv/hayama-archive/2009/coman_001.pdf related to isotropic pole and Lelong number. $\endgroup$– user21574Commented Jun 9, 2017 at 2:58
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$\begingroup$ Doesn't the lelong number calculate the coefficient in front of the log, which I have already assumed is one? I am interested in the constant term. $\endgroup$– Poincare-LelongCommented Jun 9, 2017 at 4:55
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1$\begingroup$ Yes of course, $\omega$-psh Green functions with one pole of maximal Lelong number 1 must have an isotropic pole. Are you looking for page 1 , math.leidenuniv.nl/~pbruin/scriptie.pdf ? $\endgroup$– user21574Commented Jun 9, 2017 at 5:20
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1 Answer
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This turned out to be a highly non-trivial topic. For a detailed analysis, see Jorgenson and Kramer's paper Bounds on canonical Green functions. Their result can be summarized as $$ g_{\textrm{can},X_1}-\sum_{\gamma\in S_{\Gamma_{X_1}}(\delta;z,\omega)}g_{\mathbb{H}}(z,\gamma \omega)=O_{X_0,\delta}\left(1+\frac{1}{\lambda_{X_1,1}}\right). $$ Roughly speaking, their result indicated up to a finite cover, the "canonical Green function" on the cover minus the hyperbolic Green function given by the log function is bounded inversely proportional to the first eigenvalue of the Laplacian on the covering Riemann surface. So up to a certain constant, the growth of $A$ is controlled by the reciprocal of the first eigenvalue of the Riemann Surface.
answered Jul 10, 2017 at 1:52
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$\begingroup$ Personally, I feel quite excited with the paper. It seems this set the stage for their later work on Faltings delta function and related topics. The fact that this is related to spectral geometry is also a very nice discovery. $\endgroup$ Commented Jul 11, 2017 at 4:31