Rigidity of closed CSL submanifolds in the unit sphere

Luo, Yong; Sun, Linlin

doi:10.4171/aihpc/50

Luo, Yong ; Sun, Linlin

Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 3, pp. 531-555

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Résumé

We are concerned with the rigidity of contact stationary Legendrian (CSL) submanifolds, critical points of the volume functional of Legendrian submanifolds in a Sasakian manifold, whose Euler–Lagrange equation is a third-order elliptic PDE. We obtain several optimal rigidity theorems for closed CSL submanifolds in the unit sphere by utilizing the maximum principle together with Simons’ identity. In particular, we proved that a closed CSL submanifold $M^{n} \subset 𝕊^{2 n + 1}$ is a totally geodesic sphere or a Calabi $2$ -torus if ${| 𝐁 |}^{2} \leq \frac{4 (n - 1)}{n} + \frac{3 n - 2}{n^{2}} {| 𝐇 |}^{2}$ , where $𝐁$ and $𝐇$ are the second fundamental form and the mean curvature vector, respectively. Moreover, an example shows that this assumption is optimal.

Accepté le : 2022-01-03
Publié le : 2022-09-16

DOI : 10.4171/aihpc/50

Classification : 53C24, 53C40
Keywords: Gap theorem, contact stationary Legendrian submanifolds

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     title = {Rigidity of closed {CSL} submanifolds in the unit sphere},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     doi = {10.4171/aihpc/50},
     language = {en},
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Luo, Yong; Sun, Linlin. Rigidity of closed CSL submanifolds in the unit sphere. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 3, pp. 531-555. doi: 10.4171/aihpc/50

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