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 $n^{\dim \mathcal{ML}_0(M)}$.
For every closed oriented smooth four-manifold $M$ admitting a metric of strictly negative sectional curvature, we prove that $\dim_{\mathbb{Q}}H_1(T(M);\mathbb{Q})=\dim_{\mathbb{Q}}H_1(T_{\mathrm{TOP}}(M);\mathbb{Q})=\infty$ where $T(M)$ and $T_{\mathrm{TOP}}(M)$ denote the Torelli groups. This extends a theorem of Bu...
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 $$ \max_M S>\frac{n(288\sqrt6\,n-288)}{(288\sqrt6+192)n-137}>\frac{2n}{3}+\frac{31-9\sqrt6}{75}(n-2). $$
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\...
For $n\leq 18$, we prove that any smooth immersion of the $n$-torus into the closed unit ball in $\mathbb R^q$ has a point at which the spherical average of $\lvert II(v,v)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for t...
Let $M$ be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover $\tilde M$ has Euclidean volume growth, then $\pi_1(M)$ is finitely generated and virtually $\mathbb Z^k$ for some $0\le k\le4$. This confirms a conjecture of Pan and Rong \cite{PR18} in dimension five; see also \cite{BNS25...
Let $M^n\subset\mathbb{S}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=|h|^2$ and constant $f_3=\operatorname{tr} h^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\geq2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal cu...
Jian-Quan Ge, Hui-Xin Tan, Wen-Jiao Yan et al.· 3 citations
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