论文标题
形状的线性电势和应用
Linear potentials and applications in conformal geometry
论文作者
论文摘要
在本文中，我们得出了远离薄亚集的线性电位的估计。而且，受Huber著名作品的启发，我们验证，对于某个点很薄的子集，总有一个地理位生，它可以达到该点，并避免了一般维度的薄子集。随着这些估计值在线性电位上的应用，我们考虑了标量曲率方程式，并稍微改善了Schoen-yau和Carron在Hausdorff尺寸上的奇异集尺寸的结果，这些奇数代表了在维度大于3的范围中的域中完整的整形指标的末端。 套。更有趣的是，我们基于潜在理论的方法在维度4中的Q展界方程的单数集中产生了更强的有限定理，这是Huber定理的显着类似物。
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.
