AI 中文总结
本文证明满足Dupin条件的光滑浸入欧氏超曲面局部实解析且半代数，并由此推出紧致嵌入超曲面具有紧致性。
AI 中文摘要
我们考虑满足Dupin条件的光滑浸入欧氏超曲面，其中主重数在局部上是常数。我们证明在这种情况下，每个局部图像分支都是实解析且局部Nash的，即实解析且半代数的，无需对正则轨迹作任何局部有限性假设，也无需对奇异轨迹作任何正则性假设。证明结合了关于适当Dupin片的维度相关代数次数界与使用有限射流和结式的延拓论证。此外，解析Hessian准则随后表明每个平方距离函数都是Morse-Bott的。作者早先关于紧致性的刻画因此蕴含每个满足此条件的紧致嵌入超曲面都是紧致的。
英文摘要
We prove that a smooth immersed Euclidean hypersurface is locally Nash if it satisfies the Dupin condition wherever the principal multiplicities are locally constant. No local finiteness assumption on this regular locus, or regularity assumption on its complement, is required. The proof combines a dimension-dependent degree bound on proper Dupin pieces with a continuation argument using finite jets and resultants. An analytic Hessian criterion then implies that all squared distance functions are Morse-Bott, and hence that every compact embedded hypersurface in this class is taut. In each dimension there are only finitely many such hypersurfaces up to Nash isotopy through embedded Dupin hypersurfaces. We also obtain finiteness results for taut submanifolds in arbitrary codimension and derive topological restrictions from classical classification theorems for taut embeddings.
Comments17 pages. Added finiteness and deformation results for compact embedded Dupin hypersurfaces (Theorem C) and taut submanifolds in arbitrary codimension. Exposition and references revised
