L2 harmonic forms on submanifolds in a Hadamard manifold. (September 2015) - Lin, Hezi
Abstract: In the present paper, we first show that an n − 1 n -super-stable complete minimal submanifold in R n + m admits no nontrivial L 2 harmonic 1 -forms, and has only one end, which generalized Cao–Shen–Zhu's result on stable minimal hypersurface in R n + 1 . For L 2 harmonic forms of higher order, we prove a vanishing and finiteness theorem under the assumptions on Schrödinger operators involving the squared norm of the traceless second fundamental form. Finally, if the potential function of the Schrödinger operator is a multiple of the squared norm of the second fundamental form, we can further show that the hypersurface has only one end or finitely many ends.