AI 中文总结
该研究证明了闭欧氏球中满足特定条件的自由边界高斯f-极小子流形的拓扑分类，给出严格挤压时的拓扑结果，并构造了对应旋转例的局部族。
AI 中文摘要
设$M^k\subset \overline{B}_R^N$是闭欧氏球中的光滑紧连通可定向自由边界$f_c$-极小子流形，其中$f_c(x)=c|x|^2/2$且$c\ge 0$。假设$cR^2\le k$，且$|A_{x^\perp}|^2\le 1+\frac{1}{k-1}(1-c|x^\perp|^2)^2$（$A_{x^\perp}(X,Y)=\langle x^\perp,A(X,Y)\rangle$）。我们证明$M$微分同胚于$D^k$或$S^1\times D^{k-1}$，严格挤压时为圆盘。证明用到平方距离函数的海森凸性、其极小集上的零性估计及次水平集论证；二维余一维时，非圆盘分支旋转对称。还为小$c\ge 0$构造了嵌入旋转例的局部族，$c=0$时对应临界悬链面。
英文摘要
Let $M^k\subset \overline{B}_R^N$ be a smooth compact connected orientable free boundary $f_c$-minimal submanifold of the closed Euclidean ball, where $f_c(x)=c|x|^2/2$ and $c\ge 0$. Assume that $cR^2\le k$ and $|A_{x^\perp}|^2\le 1+\frac{1}{k-1}(1-c|x^\perp|^2)^2$, where $A_{x^\perp}(X,Y)=\langle x^\perp,A(X,Y)\rangle$. We prove that $M$ is diffeomorphic either to $D^k$ or to $S^1\times D^{k-1}$; strict pinching yields the disk. The proof uses Hessian convexity of the squared-distance function, a nullity estimate along its minimum set, and a sublevel-set argument. In dimension two and codimension one, the non-disk branch is rotationally symmetric. We also construct a local family of embedded rotational examples for small $c\ge 0$, with the $c=0$ member equal to the critical catenoid.
