发表机构
Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究高余维二次收缩平移子，证明在Andrews-Baker阈值处余维约化，并在更强收缩条件下得到刚性分类：平移子为仿射平面或碗状孤子。
AI 中文摘要
我们研究了在第二基本形式满足二次收缩条件下，任意余维中平均曲率流的完整平移孤子。我们在Andrews-Baker阈值处证明了余维约化定理：若一个完备的$n$维连通平移子等距浸入$(\mathbb{R}^{n+p},\langle\cdot,\cdot\rangle)$，且满足$|B|^2\leq\left(\frac{4}{3n}-\varepsilon_{0}\right)|H|^2$，其中$\varepsilon_{0}>0$，则它要么是仿射$n$平面，要么$|H|>0$处处成立且该平移子包含于一个$(n+1)$维仿射子空间中。特别地，这改进了低维情形下由古代平均曲率流的余维约化结果所获得的收缩范围。我们的证明纯属椭圆型：将平移子视为加权极小子流形，我们将漂移拉普拉斯的Simons型恒等式与精细梯度估计及Omori-Yau极大值原理相结合。作为推论，在更强的收缩条件$|B|^2\leq(c_n-\varepsilon_{0})|H|^2$（其中$c_n=\min\left\{\frac{4}{3n},\frac{1}{n-2}\right\}$，当$n\geq3$；当$n=2$时$c_n=2/3$）下，我们得到刚性定理：平移子要么是仿射平面，要么是包含于$(n+1)$维仿射子空间中的碗状孤子。无需熵假设。
英文摘要
We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete $n$-dimensional connected translator isometrically immersed in $(\mathbb{R}^{n+p},\langle\cdot,\cdot\rangle)$ satisfies \[ |B|^2\leq \left(\frac{4}{3n}-\varepsilon_{0}\right)|H|^2 \] for some $\varepsilon_{0}>0$, then either it is an affine $n$-plane or $|H|>0$ everywhere and the translator is contained in an $(n+1)$-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition \[ |B|^2\leq (c_n-\varepsilon_{0})|H|^2,\qquad c_n=\min\left\{\frac{4}{3n},\frac{1}{n-2}\right\} \] for $n\geq3$, with $c_n=2/3$ when $n=2$, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an $(n+1)$-dimensional affine subspace. No entropy assumption is required.
Comments26 pages. Comments are welcome!
