发表机构
Chiba University（千叶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究四维伪脐双调和子流形，通过引入λ-双极小子流形，证明六维欧氏空间中此类子流形必为极小，部分回答了Chen猜想。
AI 中文摘要
1989年，Dimitrić证明了当子流形的维数不等于四时，欧氏空间中的任何伪脐双调和子流形都是极小的。然而，四维情形近四十年来一直悬而未决。这一困难源于在维数为四时，双调和方程的切向部分恒为零。因此，我们考虑λ-双极小子流形，其定义方程推广了双调和方程的法向部分。在本文中，我们证明了欧氏空间中任何连通的、余维数为二且维数至少为三的λ-双极小伪脐子流形具有常平均曲率。此外，当λ非负时，我们证明它是极小的。作为推论，我们证明了六维欧氏空间中的任何四维伪脐双调和子流形是极小的。这一结果也为Chen猜想提供了部分肯定回答。
英文摘要
In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected pseudo-umbilical 0-biminimal submanifold of codimension two and dimension at least three in a Riemannian manifold of constant sectional curvature $c$ has constant mean curvature. Furthermore, when $c$ is non-positive, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in a six-dimensional Riemannian manifold of non-positive constant sectional curvature is minimal. These results also provide partial affirmative answers to Chen's conjecture, generalized Chen's conjecture for non-positive constant sectional curvature, and the Balmuş-Montaldo-Oniciuc conjecture.
Comments34 pages; the title was changed; Theorem 1.3 was added
