  In this paper we derive estimates for linear potentials that hold away from
thin subsets. And, inspired by the celebrated work of Huber, we verify that,
for a subset that is thin at a point, there is always a geodesic that reaches
to the point and avoids the thin subset in general dimensions. As applications
of these estimates on linear potentials, we consider the scalar curvature
equations and slightly improve the results of Schoen-Yau and Carron on the
Hausdorff dimensions of singular sets which represent the ends of complete
conformal metrics on domains in manifolds of dimensions greater than 3. We also
study Q-curvature equations in dimensions greater than 4 and obtain stronger
results on the Hausdorff dimensions of the singular sets. More interestingly,
our approach based on potential theory yields a significantly stronger
finiteness theorem on the singular sets for Q-curvature equations in dimension
4, which is a remarkable analogue of Huber's theorem.
