AI 中文总结
本文研究球面中闭连通非全测地极小子流形的体积间隙问题，给出了带重数点相关的体积下界、线性满浸入时的体积下界，以及超平面截得的连通分支体积下界与分支数上界，推进了极小子流形体积估计的研究。
AI 中文摘要
设$f:M^n\to\bS^{n+q}(1)$（$n\bge2$且$q\bge1$）为闭的连通非全测地极小浸入，第二基本形式为$h$，记$S=|h|^2$且$S_*=\bmax_M S$。若$p\bim f(M)$的重数为$m$且$f^{-1}(p)=\brace{x_1,\bdots,x_m}$，则$\bVol(M)\bge\bl[m+\bvarepsilon_n\bsum_{j=1}^m\bl(\frac{S(x_j)}{S_*}\br)^2\br]\bVol(\bS^n)$，其中$[110n(n+2)^2]^{-1}<\bvarepsilon_n<[104n(n+2)^2]^{-1}$。若浸入是线性满的，则$\frac{\bVol(M)}{\bVol(\bS^n)}\bge\bmax\bcbrace{1+\bvarepsilon_n,\frac{4(n+1)^n}{(n+3)^{n+2}}(n+q+1)}$。此外，对每个过原点的超平面$H$，$M\bsetminus f^{-1}(H)$的每个连通分支体积至少为$4(n+1)^n(n+3)^{-n-2}\bVol(\bS^n)$，因此分支数至多为$\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\bVol(M)}{\bVol(\bS^n)}$。
英文摘要
Let $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ and $q\ge1$, be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form $h$, and put $S=|h|^2$ and $S_*=\max_M S$. If $p\in f(M)$ has multiplicity $m$ and $f^{-1}(p)=\{x_1,\ldots,x_m\}$, then \[ \Vol(M)\ge \left[m+\varepsilon_n\sum_{j=1}^m \left(\frac{S(x_j)}{S_*}\right)^2\right]\Vol(\Sph^n), \] where $[110n(n+2)^2]^{-1}<\varepsilon_n<[104n(n+2)^2]^{-1}$. If the immersion is linearly full, then \[ \frac{\Vol(M)}{\Vol(\Sph^n)} \ge \max\!\left\{1+\varepsilon_n, \frac{4(n+1)^n}{(n+3)^{n+2}}(n+q+1)\right\}. \] Moreover, for every hyperplane $H$ through the origin, each connected component of $M\setminus f^{-1}(H)$ has volume at least $4(n+1)^n(n+3)^{-n-2}\Vol(\Sph^n)$; consequently the number of components is at most $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.
Comments28 pages. Comments welcome
