# What condition needs to be placed on a manifold for integration by parts to not have boundary term at infinity?

Say you have a complete, noncompact Riemannian manifold without boundary $$(M,g)$$. I am wondering what condition needs to be placed on the metric $$g$$ so that when you perform an integration by parts, it simplifies to $$\int_M v \Delta u\,dV=-\int_M \langle\nabla v,\nabla u\rangle\,dV$$ That is to say, I am looking for a condition on the metric so that the `boundary' integral term at infinity vanishes.

I was trying to think about this problem using an auxiliary function $$\eta$$ that has compact support (say in $$B_{R+1}(p)$$) and $$\eta\equiv 1$$ in $$B_R(p)$$. Then multiplying the integrand by $$\eta$$ and integrating by parts (since the integrand has compact support so the boundary term vanishes), you're left with exactly what an error term of the form $$\int_{B_{R+1}\setminus B_R} v\langle\nabla \eta, \nabla u\rangle \,dV$$ but I am not sure exactly what condition I need to specify to have this term go to 0 as $$R\to \infty$$.

For instance, is the condition that $$(M,g)$$ has bounded geometry (ie bounded curvature and positive injectivity radius) sufficient? Or does anyone have a reference for this?

• You need conditions on $u$ and $v$, not merely on the metric. Given any noncompact Riemannian manifold, one can always find functions which are badly behaved at infinity. May 14 at 2:14
• @Kajelad sure I totally agree. I’m assuming all these integrals exist at the very least. I’d be willing to assume $u,v\in W^{2,2}$ May 14 at 3:06
• I'm not sure that Sobolev-type conditions are sufficient. In $\mathbb{R}^n$, for instance, it's common to use falloff conditions such as $$\exists C>0\text{ s.t. }\|\nabla^ku\|(x)<C\|x\|^{-p}$$Things are more complicated in the general case, but you could probably write something similar e.g. by pulling back by the exponential map. May 14 at 6:10