Hello,

On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like $$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\epsilon^{-1}\lVert u\rVert_{L^2}^2$$ for $u \in H^2,$ where $\epsilon$ is of my choosing.

Does this or something similar hold (eg. instead, the inequality can have the LHS in $H^1$ norm and the $H^1$ norm on the RHS can be the $H^2$ norm.)

I saw something like this in Adams but this requires the domain to satisfy some unpleasant conditions. Thanks.