  We obtain a fundamental gap estimate for classes of bounded domains with
quantitative control on the boundary in a complete manifold with integral
bounds on the negative part of the Ricci curvature. This extends the result of
\cite{Oden-Sung-Wang99} to $L^p$-Ricci curvature assumptions, $p>n/2$. To
achieve our result, it is shown that the domains under consideration are John
domains, what enables us to obtain an estimate on the first nonzero Neumann
eigenvalue, which is of independent interest.
