Aug 2026· 2 citations· ⚡ 1 influential· 29 references
Mathematics
Abstract
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 \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \]
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\...
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}^...
For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Eu...
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,\math...
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 scal...
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 vo...
