A topological sphere theorem for pinched Legendrian submanifolds

Abstract

Let $M^n$ be a closed simply connected Legendrian submanifold of $\mathbb S^{2n+1}$, where $n\geq5$, and let $\mathbf{B}$ and $\mathbf{H}=\operatorname{tr}\mathbf{B}$ denote its second fundamental form and mean-curvature vector. We prove that $M$ is homeomorphic to $\mathbb S^n$ if \begin{equation*} |\mathbf{B}|^2\leq \frac{(n-1)(n+2)}n+\frac{n^2+3n-2}{2n^2}|\mathbf{H}|^2-\frac{(n-1)(n-2)}{2n^2}|\mathbf{H}|\sqrt{|\mathbf{H}|^2+4n}. \end{equation*} The equality can be attained by the Calabi products, so simple connectivity cannot be omitted. For $n\geq4$, we also prove a sphere theorem under an explicit bound and the weaker assumption $b_1(M;\mathbb R)=0$. The proof combines phase-rotated Lawson--Simons variations with a systolic inequality for finite two-complexes. We also give local constructions showing that the pinching condition allows negative isotropic curvature at a point.