发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二维球面等距浸入的退化Weyl问题，在仅非负高斯曲率条件下，分别给出了全局$C^{2,1}$嵌入和$W^{3,p}$浸入的存在性结果，并解决了非退化情形。
AI 中文摘要
本文关注Weyl问题，即二维球面到三维欧几里得空间或一般环境三维流形的等距浸入或嵌入的存在性。我们在退化Weyl问题上建立了两个结果，即当高斯曲率$K_g$仅为非负而非严格正时。首先，对于二维球面上的光滑度量$g$，若$K_g$除有限个点外严格为正，且在这些点上$K_g$的Hessian正定，则$g$容许一个全局$C^{2,1}$-等距嵌入到$\mathbb{R}^3$中。这似乎是退化Weyl问题中首个仅基于$g$的内在条件的结果。其次，对于一般单连通环境三维流形$(\mathcal{M},{\overline{g}})$，若对某常数$K_0$有$K_g \geq K_0 \geq {\rm sec}_{\overline{g}}$，$(K_g-K_0)^{-1/2} \in L^p$（$p \geq 2$），且对近似非退化等距浸入满足某种一致夹逼条件，则存在一个$W^{3,p}$-等距浸入。同时，我们还解决了$g$、${\overline{g}} \in C^{2,1}$时到一般单连通环境三维流形的非退化Weyl问题（即当$K_g>0$时）。
英文摘要
This paper is concerned with the Weyl problem, \emph{i.e.}, the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature $K_g$ is only nonnegative rather than strictly positive. First, for a smooth metric $g$ on the two-sphere, if $K_g$ is strictly positive except at finitely many points where the Hessian of $K_g$ is positive definite, then $g$ admits a global $C^{2,1}$-isometric embedding into $\mathbb{R}^3$. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on $g$. Second, for a general simply-connected ambient three-manifold $(\mathcal{M},{\overline{g}})$, if $K_g \geq K_0 \geq {\rm sec}_{\overline{g}}$ for some constant $K_0$, $(K_g-K_0)^{-1/2} \in L^p$ with $p \geq 2$, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a $W^{3,p}$-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (\emph{i.e.}, when $K_g>0$) into general simply-connected ambient three-manifolds for $g$, ${\overline{g}} \in C^{2,1}$.
