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 $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.
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...
Let $F:M^n\to\Sn^{n+q}(1)$ be a closed minimal immersion in the unit sphere, with $n\ge3$, $q\ge2$. Let $S$ be the squared length of its second fundamental form. We prove that if $M$ is not totally geodesic, then $$ \max_M S>\frac{n(288\sqrt6\,n-288)}{(288\sqrt6+192)n-137}>\frac{2n}{3}+\frac{31-9\sqrt6}{75}(n-2). $$
Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequentia...
Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R<\infty$, $$c_i\l...
Let $n,m$ be integers such that $n\geq2$ and $0\leq m\leq n-2$. Let $C_{m+1}$ denote the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove that there are constants $\nu(n,m),C(n,m)>0$ such that the following holds. If $(M^n,g)$ is complete and connected and, for $\delta\geq 0$, \[ \mathrm{Ri...
Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral hom...
Yi-Fan Zhang· 0 citations
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