We derive estimates for linear potentials that hold away from thin subsets. And,
inspired by the celebrated work of Huber (1957) and Cohn-Vossen (1935), 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 improve
the results of Schoen and Yau (1988, 1994) and Carron (2012) on the Hausdorff
dimensions of singular sets which represent the ends of complete conformal
metrics on domains in manifolds of dimension greater than 3. We also study
-curvature
equations in dimensions greater than 4 and obtain stronger results on
the Hausdorff dimensions of the singular sets than those of Chang et
al. (2004). More interestingly, our approach based on potential theory
yields a significantly stronger finiteness theorem on the singular sets for
-curvature
equations in dimension 4 than those of Chang et al. (2000) and Carron and Herzlich
(2002), which is a remarkable analogue of Huber’s theorem.
