Symmetry and compact embeddings for critical exponents in metric-measure spaces

Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral approximation.

Covering maps and ideal embeddings of compact homogeneous spaces

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) is an isometrical embedding which receives the least possible amount of tension from the surrounding space at each point. In this article, we study ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces via covering maps. Our main result states that $π: M\to N$ is a c…

等距嵌入共形紧致流形到双曲空间

Isometric Embeddings of Conformally Compact Manifolds into Hyperbolic Spaces

本文证明了共形紧致流形可以等距嵌入到高维双曲空间中，解决了在无穷远处具有负曲率下界流形的嵌入问题。

Uniqueness of the stereographic embedding

The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.

Embeddings of compact Sasakian manifolds

Let M be a compact Sasakian manifold. We show that M admits a CR-embedding into a Sasakian manifold diffeomorphic to a sphere, and this embedding is compatible with the respective Reeb fields. We argue that a stronger embedding theorem cannot be obtained. We use an extension theorem for Kaehler geometry: given a compact Kaehler manifolds $X\subset Y$, and a Kaehler form $ω$ on $X$ which lies in a Kaehler class of $Y$ restricted to $X$, $ω$ can be extended to a Kaehler form on $Y$.

Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary

In this paper, using blow-up analysis, we prove a quantization result for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary. Similar results for Euclidean space were obtained by Adimurthi-Struwe \cite{Adi-Stru}, Druet \cite{Druet}, Lamm-Robert-Struwe \cite{L-R-S}, Martinazzi \cite{Mart}, Martinazzi-Struwe \cite{Mar-Stru}, and Struwe \cite{Struwe} respectively.