A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.
Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \]
Let $M^n$ be a closed minimal submanifold in the unit sphere $\mathbb{S}^{n+q}$ with $n\geqslant 3$ and $q\geqslant 2$. Let $S$ be the squared length of its second fundamental form and $S_{\max}=\max_{p\in M}S(p)$. We prove that if $M$ is not totally geodesic, then \[ S_{\max}>\frac{2n}{3}+\frac{n-2}{182}. \]
We prove that if the $n$-dimensional torus $T^n$ is smoothly immersed in the unit ball $B^q\subset\mathbb R^q$ and $n\leq 18$, then there exists a point at which the spherical average of $\lvert II(u,u)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar-curvature obstruction for the torus with a conformal-Laplacian argument.
We show that in any Haken 3-manifold $M$, the dimensions of $\mathcal{ML}(M)$ and $\mathcal{ML}_0(M)$ can be calculated using normal surfaces. As a corollary, we prove the number of closed orientable essential surfaces of Euler characteristic at least $-2n$ in a closed orientable hyperbolic 3-manifold with $H_2(M,\mathbb{Z}/2\mathbb{Z}) = 0$ is asymptotic to $n^{\dim \mathcal{ML}_0(M)}$.
Let $M^n$ $(n\geqslant3)$ be a closed minimal submanifold in the unit sphere $\mathbb S^{n+m}$ $(m\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and \[ 0\leqslant S\leqslant n+\delta, \] where $\delta$ is an explicit constant satisfying $\delta\geqslant \frac{n}{87}$, then either $S\equiv0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\mathbb S^{n+1}\subset\mathbb S^{n+m}$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern's conjecture in higher codimension.
Jianquan Ge, Fagui Li, Yunheng Zhang· 7 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
