There are all sorts of kinds of monotonicity for various PDEs. Generally, one uses monotonicity together with an $\epsilon$-regularity result to get strong restriction on the structure of the singularity set. The most basic monotonicity results take the form $E(f, S_r) \le CE(f, S_R)$ for $E$ some (renormalized) energy of a solution $f$ to some PDE on spheres of radius $r < R$ and $C$ not dependent on $r, R$. A monotonicity of this sort exists for harmonic maps ($d^* d f = 0$) with $E(f,\Omega) = r^{2-n} \|df\|_{L^2(\Omega)}^2$ and Yang-Mills connections $(d_A^* F_A = 0$) with $E(A, \Omega) = r^{4-n} \|F_A\|^2_{L^2(\Omega)}$.

Mark showed me this one. We start off with integrating a Bochner-type formula to get $$\int_{B_R} \Delta \frac{|h|^2}{2} \mathrm{vol}_{B_R} = \int_{B_R} (|\nabla h|^2 + \langle \mathcal{R}h, h\rangle) \mathrm{vol}_{B_R},$$ with $\mathcal{R}$ a curvature term. By the divergence theorem, the left-hand side is $$\int_{B_R} \Delta \frac{|h|^2}{2} = \int_{S_R} \partial_r \frac{|h|^2}{2} \mathrm{vol}_{S_R} \approx R^{n-1} \frac{\partial}{\partial r} \left( R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R} \right).$$ The last approximation is exact in Euclidean space, but holds to first order by working in geodesic coordinates. Thus we have $$\frac{\partial}{\partial r} \left(R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R} \right) \approx R^{1-n} \int_{B_R} (|\nabla h|^2 + \langle \mathcal{R}h, h\rangle) \mathrm{vol}_{B_R},$$ which establishes monotonicity for the energy $R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R}$ if we know that the approximation is good enough and the curvature is positive enough to make the integrand of the right hand side positive.