除测地线、极小曲面和极小子流形外，面积最小化子流形一般非光滑
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
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中文总结 AI 辅助
研究模2同调中面积最小化子流形的光滑性，证明其一般非光滑（除特定情况），解决怀特猜想，还建立奇异集豪斯多夫维数下界，并证明$\rpt$维罗内塞极小嵌入上的锥是模2面积最小化的。
中文摘要 AI 辅助
我们证明了在模2同调中面积最小化子流形一般非光滑，除了测地线、极小曲面和极小子流形的情况。这解决了怀特关于模2同调中面积最小化子流形一般光滑性的猜想。此外，我们建立了关于黎曼度量开集的面积最小化子流形奇异集的豪斯多夫维数的下界。下界是$(d - 3)$，其中$d$表示子流形的维数。作为关键步骤，我们证明了$\rpt$的维罗内塞极小嵌入上的锥是模2面积最小化的，解决了另一个长期存在的开放问题。
英文摘要
We prove that area-minimizing submanifolds in mod $2$ homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This answers questions of White and Yau that ask about the generic smoothness of area-minimizing submanifolds. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is ${(d-3)}$ where $d$ denotes the dimension of the submanifold. As a crucial step, we prove that the cone over the Veronese minimal embedding of $\mathbb{RP}^2$ is mod $2$ area-minimizing, settling another long-standing open problem.
