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.
In this paper, we establish a general volume estimate for complete Riemannian manifolds under suitable differential inequalities involving a proper function and a symmetric tensor, without imposing curvature assumptions. As an application, we prove that a class of orientable hypersurfaces properly immersed in Euclidean...
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 construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo con...
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...
For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed...
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...
Brevan Ellefsen· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
