laplacian of Busemann function on hyperbolic space [closed]

M is a hyperbolic space $\mathbb{H}^n.$ $secM=-1.$ $γ(t) : R \rightarrow M$ a line.Let $b_+$ be the Busemann function for $γ : [0,\infty) \rightarrow M,$ and $b_−$ the Busemann function for $γ : (−\infty, 0] \rightarrow M.$

Thus, $b_+(x) = \lim_{t\rightarrow \infty}(d(x, γ(t)) − t),b_−(x) = \lim_{t\rightarrow \infty}(d(x, γ(−t)) − t).$ I want to compute $\Delta b_+,\Delta d_-.$

Below is my computation,I don't know whether it's right. Δb+<=limt→+∞(n-1)cotht=n-1 Δb-<=limt→-∞(n-1)cotht=-(n-1) so Δ(b++b−)<=0,and (b++b−)(γ(t))=0,(b++b−)(M)>=0, so can I claim (b++b−)(M)=0? (Ric>=0,we can use mean value property for supharmonic function to prove the claim,but now Ric=-(n-1)) then get Δb+=n-1,Δb-=-(n-1).

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 Please, use TeX when asking question, since your question is almost unreadable. Your argument will work only for 1-dimensional hyperbolic space. – Misha Oct 15 at 12:22 I am voting to close this question until it is typeset properly. – Igor Rivin Oct 15 at 12:37

closed as too localized by Deane Yang, Misha, Igor Rivin, Alain Valette, AgolOct 15 at 17:07

If memory serves, this computation is done in:

Besson, G.(F-GREN-F); Courtois, G.(F-GREN-F); Gallot, S.(F-GREN-F) Volume et entropie minimale des espaces localement symétriques. (French) [Volume and minimal entropy of locally symmetric spaces] Invent. Math. 103 (1991), no. 2, 417–445.

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Thank you!I am quite sorry that I don't know how to use Latex.Maybe my question can be simplified:What's the laplacian of the Buseman function on Hyperbolic space H^n?When restricted to geodesic ,is it linear?And the level sets are totally geodesic? – jiangsaiyin Oct 15 at 12:56
Level sets are horospheres. – Igor Rivin Oct 15 at 14:14
by the way,the paper you recommend is written in French,I cannot read it. – jiangsaiyin Oct 15 at 14:57
Any serious student of Riemannian geometry should be able to read french papers, too. – Igor Rivin Oct 15 at 16:04
@R W you did insult me, together with @Agol @Deane Yang @Misha and @Alain Valette -- all of us are known to know a little something about hyperbolic geometry, and all of us are telling the MathOverflow world who we are. An anonymous person who makes incendiary or insulting remarks on discussion boards is a troll in my book. – Igor Rivin Oct 16 at 23:40