On Huber's type theorems in general dimensions

Gromov-Hausdorff convergence theory of surfaces

In this paper, we use the viewpoint of Gromov-Haustorff convergence to give some new comprehension of well known theorem,it is Huber's classification theorem\cite{Huber}\cite{MS}for complete Riemannian surfaces immersed in $\mathbb{R}^n$ with finite total curvature( $\int_Σ|A|^2<+\infty$) it depend heavily on Müller and Šverák's Hardy-estimate\cite{MS} for the curvature form of surfaces immersed in $\mathbb{R}^n$ with finite total curvature.

On sufficient conditions to extend Huber's finite connectivity theorem to higher dimensions

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite total curvature where $n \geq 2$. Note here that our radial curvatures can change signs wildly. We then show that $\lim_{t\to\infty} \mathrm{vol} B_t(p) / t^n$ exists where $\mathrm{vol} B_t(p)$ denotes the volume of the open metric ball $B_t(p)$ with center $p$ and radius $t$. Moreover we show that in addition if the limit above is positive, then $M$ has finit…

Linear potentials and applications in conformal geometry

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 th…

奇异CR Yamabe问题与Hausdorff维数

The Singular CR Yamabe Problem and Hausdorff Dimension

本文研究紧致伪厄米流形上接触形式共形形变产生的奇异CR Yamabe问题，估计奇异集的Hausdorff维数，并基于正质量定理证明三维CR发展映射的单射性。

无共轭点的亏格至少为2的曲面上的测地环计数

Counting geodesic loops on surfaces of genus at least 2 without conjugate points

本文在无共轭点的紧曲面上建立了闭测地环的渐近计数估计，并证明了更一般的扇形定理，推广了Huber、Margulis、Nicholls和Sharp等人在严格负曲率情形下的经典结果。

势论与共形几何中的应用

Potential theory and applications in conformal geometry

本文探讨了势论在共形几何中的应用，建立了Huber型定理和端部的Hausdorff维数估计。

一般Kastler-Kalau-Walze型定理与带边界的奇数维流形上的Dabrowski-Sitarz-Zalecki型定理

The general Kastler-Kalau-Walze type theorem and the Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifold with boundary

本文证明了带边界的奇数维紧流形上的一般Kastler-Kalau-Walze型定理和Dabrowski-Sitarz-Zalecki型定理。

The Higher Dimensional Positive Mass Theorem II

We derive the Space-Time Positive Mass theorem in arbitrary dimensions, without topological constraints. The main new tools are skin structures and surgeries on minimal and marginally outer trapped hypersurfaces.

A one-sentence proof of the Bernstein type theorem for special Lagrangian equation in two dimensions

The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.

Two nontrivial index theorems in odd dimensions

A recent anomaly computation of Horava and Witten is proved and generalized in the form of two index theorems in odd dimensions. Theorem A is a fixed point formula for orientation-reversing involutions. Theorem B is an index theorem for manifolds with boundary using local boundary conditions. Both hold for families of operators.

The Higher Dimensional Positive Mass Theorem I

We derive the Riemannian Positive Mass theorem in arbitrary dimensions, without any topological constraints. The main new tools are skin structures and surgeries on minimal hypersurfaces.