Let $X$ be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that $X$ admits no complete metric with uniformly positive scalar curvature when either infinite relative $K$-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset...
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...
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...
For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $\kappa(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $\kap...
Tsz-Kiu Aaron Chow, Jingbo Wan· 0 citations
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