发表机构
Beijing Institute of Technology（北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了欧氏空间中常数量曲率闭连通浸入超曲面的球面定理，结合Ros刚性论证与填充定理，得出两凸超曲面为球面，并推广至高维刚性结果。
AI 中文摘要
我们证明了欧几里得空间中具有常数量曲率的闭连通浸入超曲面的球面定理。结合Ros的刚性论证与Huisken和Sinestrari的填充定理，我们证明了每个两凸的此类超曲面都是标准球面。作为推论，Yau猜想在\\(\mathbb R^4\\)中的闭连通浸入超曲面上成立。利用Gromov的填充定理，我们还在主曲率指标条件和显式数量曲率夹紧条件下获得了高维刚性结果。
英文摘要
We prove sphere theorems for closed connected immersed hypersurfaces of constant scalar curvature in Euclidean space. Combining Ros's rigidity argument with the filling theorem of Huisken and Sinestrari, we show that every two-convex such hypersurface is a round sphere. As a consequence, Yau's conjecture holds for closed connected immersed hypersurfaces in \(\mathbb R^4\). Using Gromov's filling theorem, we also obtain higher-dimensional rigidity results under a principal-curvature index condition and an explicit scalar-curvature pinching condition.
Comments14 pages
