Geodesic
VANISHING THEOREMS FOR HYPERSURFACES IN THE UNIT SPHERE
Glasgow mathematical journal, Tome 60 (2018) no. 3, pp. 661-671

Voir la notice de l'article provenant de la source Cambridge University Press

Let Mn, n ≥ 3, be a complete hypersurface in $\mathbb{S}$n+1. When Mn is compact, we show that Mn is a homology sphere if the squared norm of its traceless second fundamental form is less than $\frac{2(n-1)}{n}$. When Mn is non-compact, we show that there are no non-trivial L2 harmonic p-forms, 1 ≤ p ≤ n − 1, on Mn under pointwise condition. We also show the non-existence of L2 harmonic 1-forms on Mn provided that Mn is minimal and $\frac{n-1}{n}$-stable. This implies that Mn has only one end. Finally, we prove that there exists an explicit positive constant C such that if the total curvature of Mn is less than C, then there are no non-trivial L2 harmonic p-forms on Mn for all 1 ≤ p ≤ n − 1. 
LIN, HEZI. VANISHING THEOREMS FOR HYPERSURFACES IN THE UNIT SPHERE. Glasgow mathematical journal, Tome 60 (2018) no. 3, pp. 661-671. doi: 10.1017/S0017089517000350
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