Let $M^n\to\Sph^{n+q}(1)$, $n\ge3$ and $q\ge2$, be a closed minimal immersion. Set $S=|h|^2$ and $Q=S+\lambda_2$, where $h$ is the second fundamental form and $\lambda_2$ is the second largest eigenvalue of Lu's fundamental matrix. We establish two complementary results concerning Lu's second-gap conjecture. First, for every $n\ge3$, we exhibit closed connected homogeneous minimal embeddings of $\Sph^1\times\Sph^{n-1}$ into $\Sph^{2n+1}(1)$ with constant $S$ and constant $Q$, whose $Q$-values are dense in $(n,2n)$. Totally geodesic inclusions yield the same density in every codimension $q\ge n+1$. Thus Lu's conjecture fails in these codimensions even under constant scalar curvature. Second, for every $n\ge3$, we prove that there exists $\gamma_n>0$, depending only on $n$, such that no closed connected minimal immersion into $\Sph^{n+2}(1)$ with constant $Q$ satisfies $n
Let $X: M^n\to \mathbb S^{n+1}(1)$ be a closed minimal hypersurface with constant scalar curvature. Chern conjecture says that a closed minimal hypersurface with constant scalar curvature, $S\geq 2n$ if $S>n$, where $S$ is the squared norm of the second fundamental form. As a partial result, Peng-Terng \cite{pt1, pt2},...
Qing-Ming Cheng, Feng-Jiang Li, G. Wei· 0 citations
Let $x:M^n\to\mathbb{S}^{n+q}$ be a minimal immersion of a complete connected manifold with flat normal bundle, where $n\ge3$ and $q\ge2$. We prove that if the squared norm of the second fundamental form $S$ satisfies $S<2n$, then $M^n$ lies in $\mathbb{S}^{n+1}$. The standard minimal products of three spheres show tha...
For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only...
Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $...
Oliver Janzer, István Tomon, Fredy Yip· 1 citation
For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metr...
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
