Question

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).

Answer 1

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.