A sharp pinching theorem for closed minimal Legendrian submanifolds

Abstract

Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of the minimal Calabi product of a point and a totally geodesic Legendrian sphere. This proves Conjecture 1 of Luo and Sun [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 40 (2023), p. 548].