Abstract
Let M n be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., \({\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}\), in an (n + p)-dimensional simply connected space form N n+p(c) of constant curvature c, where |H| and |A|2 are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and \({\int_M|H|^n < \infty}\); (ii) n ≥ 5, c = −1 and \({|H| < 1-\frac{2}{\sqrt n}}\); (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L 2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.
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Dedicated to Professor Shing-Tung Yau on the occasion of his 60th birthday.
Research supported by the NSFC, Grant No. 10771187, 10671087; the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China; the Natural Science Foundation of Zhejiang Province, Grant No. 101037; and the NSFJP, Grant No. 2008GZS0060.
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Fu, HP., Xu, HW. Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms. Geom Dedicata 144, 129–140 (2010). https://doi.org/10.1007/s10711-009-9392-z
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DOI: https://doi.org/10.1007/s10711-009-9392-z