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Sphere Theorems for Immersed Hypersurfaces with Constant Scalar Curvature
Abstract
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 ℝ4. 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.
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BibTeX
@misc{li2026sphere,
title = {{Sphere Theorems for Immersed Hypersurfaces with Constant Scalar Curvature}},
author = {Fagui Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29673},
eprint = {2609.29673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29673},
}
RIS
TY - UNPB
TI - Sphere Theorems for Immersed Hypersurfaces with Constant Scalar Curvature
AU - Fagui Li
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29673
UR - https://arxiv.org/abs/2609.29673
AB - 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 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.
ER -