In the Laplacian comparision theorem by Yau and Schoen, the proof assume a $n$ dimensional space form $N$ with sectional curvature $-k^{2}$. And we have a separate $M$ with $Ric<-(n-1)k^{2}$. But after reading the proof several times and scratching my head for hours, I do not understand what is the role of $N$ at here. I tried to find online sources on the same theorem, and the ones I found did not state $N$ at all. Further in the proof I noticed the authors assumed $M$ has curvature $-k^{2}$ instead. Is there a typo here or somewhere? I am really confused. This is in page 5 of Yau and Schoen (sorry I do not know where to find an online version). 


Their original theorem statement is the following:


Let $M$ be an $n$-dimensional complete Riemannian manifold with $Ric(M)\ge -(n-1)k^{2},(k\ge 0)$. Let $N$ be the $n$-dimensional simply connected space of constant sectional curvature $-k^{2}$. Let $p\_{M}$ and $p\_{N}$ be distance functions with respect to some fixed point $x$ on $M$ and $N$ respectively. If $x\in M$ and $p\_{M}$ is differentiable at $x$, then for any $y\in N$ with $p\_{N}(y)=p\_{M}(x)$ we have
$$\Delta p\_{M}(x)\le \Delta p\_{N}(y)$$


But reading their proof I could not find where did they use the assumption $Ric(M)\ge -(n-1)k^{2}$ or $N$ is simply connected at all. 



Confusion with the Laplacian comparision theorem
The form $-k^2$ is just a way of saying that the sectional curvature is nonpositive. There is a nice, modern and detailed discussion of this Laplacian comparison result in P. Petersen, "Riemannian Geometry" (second edition), Springer-Verlag, 2006. The result you want is the first inequality in Lemma 34, page 268. There, one only assumes a lower bound on the Ricci curvature of the form $Ric(M)\geq (n-1)k$, with $k\in\mathbb{R}$, and no simple connectedness assumption is made, as Anton Petrunin commented above. The proof is entirely based on Proposition 39, pages 266-267, which relates the volume form of $(M,g)$ to the Laplacian of $p\_M$. The needed inequality in that Proposition is obtained from direct calculations and the Cauchy-Schwarz inequality for the Hilbert-Schmidt scalar product of matrices.


