I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition:

Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, and the standard metric $\bar{g}$ on the sphere $\mathbb{S}^n$ restricted to the hemisphere, we define another metric $g(t)$ on the upper hemisphere by:

$g(t) = g_0(t) + \frac{1}{2(n-1)}t^2 u \bar{g}$

In the paper they say this implies:

$R_{g(t)}=R_{g_0(t)} - \frac{1}{2} t^2 (\Delta u + nu) + O(t^3)$

I'm not sure if I should break down and calculate like mad, or if there is a better way to see this. The conditions we have on $u$ are simply

$u|_{\partial \mathbb{S}^n_+}= 0$

I appreciate all help. Cheers!