Maybe my question is a nonsense but I prefer to ask.
Take $M$ a manifold that is not complete (ie the exponential is not defined for all values of $TM$) and $V$ a potential defined on such manifold.
For example : take the cylinder $S^{d-1} \times [0,1]$ and a potential $V(\theta,t) = |\theta|^2 + t^2$.
Is there a result like Bakry/Emery's one, ensuring a log Sobolev inequality ? I assume that there are conditions on the Ricci curvature and the hessian of the potential. Maybe something is happening at the boundary...
Thanks in advance !