AI 中文总结
本文研究空间形式中作为梯度 Yamabe 孤立子的完备浸入超曲面，通过分析正则水平集刻画其结构，证明其满足的分解条件与下界，明确下界达到的等距条件。
AI 中文摘要
本文研究空间形式 $\left(\overline{M}^n(C), \overline{g}\right)$ 中具有常数标量曲率的完备浸入超曲面所对应的非平凡 Yamabe 梯度孤立子。对于梯度非零的孤立子，我们通过分析孤立子函数的正则水平集 $\Sigma$ 建立了孤立子的结构刻画。特别地，对 $\Sigma$ 作为空间形式子流形的第二基本形式无迹部分 $\Phi^{\Sigma}_{\overline{M}}$ 施加合适条件后，我们证明该孤立子要么分解为 $\mathbb{R}$ 与空间形式的全脐子流形的黎曼积，要么满足 $\sup\limits_\Sigma\left|\Phi^{\Sigma}_{\overline{M}}\right|$ 的严格下界。此外，我们证明该下界达到当且仅当孤立子等距于 $\mathbb{R}$ 与空间形式的平行子流形的黎曼积。
英文摘要
This paper studies nontrivial Yamabe gradient solitons occurring as complete immersed hypersurfaces with constant scalar curvature in space forms $\left(\overline{M}^n(C), \overline{g}\right)$. For solitons with non-vanishing gradients, we establish a structural characterization of the soliton by analyzing a regular level set $Σ$ of the soliton function. In particular, imposing a suitable condition on the traceless part of the second fundamental form, $Φ^Σ_{\overline{M}}$ of $Σ$ as a submanifold of the space form, we prove that the soliton either decomposes as a Riemannian product of $\mathbb{R}$ and a totally umbilical submanifold of the space form, or satisfies a sharp lower bound on $\sup\limits_Σ\left|Φ^Σ_{\overline{M}}\right|$. Furthermore, we show that the lower bound is attained if, and only if, the soliton is isometric to a Riemannian product of $\mathbb{R}$ and a parallel submanifold of the space form.
Comments15 pages; Comments are welcome
