Global pinching theorems for minimal submanifolds in a complex projective space

In this paper, we investigate the compact minimal submanifold $M^n (n \geq 3)$ in the complex projective space $\mathbb{C}P^{\frac{n+p}{2}} (1)$. Denote by $S$ the square norm of the second fundamental form. Set $\tilde{S} = S+ \frac{1}{2} {\lvert t \rvert}^2$ and $\epsilon = \inf+M {\lvert P \rvert}^2$, where $P, t$ are the tensors defined in (2.1) below. We first prove such that if ${\lVert S \rVert}_{n/2} \lt \alpha_1 (n, \epsilon)$ for $\epsilon \gt 0$, then $M$ is a totally geodesic submanifold $\mathbb{C}P^{\frac{n}{2}}$. Moreover, we prove if ${\lVert \tilde{S} \rVert}_{n/2}\lt \alpha_3 (n)$, where $\alpha_3 (n)$ is an explicit positive constant depending only on $n$, then $M$ is a totally geodesic submanifold $\mathbb{C}P^{\frac{n}{2}}$. We also prove other global pinching theorems for minimal submanifolds in the complex projective space.

Keywords

minimal submanifold, complex projective space, second fundamental form, global pinching

2010 Mathematics Subject Classification

53C24, 53C42

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Research supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 12071424.

Received 25 January 2019

Accepted 31 July 2020

Published 15 October 2021