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In the present note, we deal with $L^2$ harmonic $1$-forms on complete submanifolds with weighted Poincar\'{e} inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^2$ harmonic $1$-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and Vit\'{o}rio.
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참고문헌 (24)
참고문헌 신청
D. M. Calderbank / 2000 / Refined Kato inequalities and conformal weights in Riemannian geometry / J. Funct. Anal 173(1):214~255
G. Carron / 1999 / L 2-Cohomologie et in´egalit´es de Sobolev / Math. Ann 314(4):613~639
M. P. Cavalcante / 2014 / L 2 harmonic 1-form on submanifolds with finite total curvature / J. Geom. Anal 24(1):205~222
X. Cheng / 2000 / L 2 harmonic forms and stability of hypersurfaces with constant mean curvature / Bol. Soc. Brasil. Mat. (N.S.) 31(2):225~239
N. T. Dung / 2012 / Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature / Ann. Global Anal. Geom 41(4):447~460
G. Carron / 1999 / L 2-Cohomologie et in´egalit´es de Sobolev / Math. Ann 314(4):613~639
M. P. Cavalcante / 2014 / L 2 harmonic 1-form on submanifolds with finite total curvature / J. Geom. Anal 24(1):205~222
X. Cheng / 2000 / L 2 harmonic forms and stability of hypersurfaces with constant mean curvature / Bol. Soc. Brasil. Mat. (N.S.) 31(2):225~239
N. T. Dung / 2012 / Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature / Ann. Global Anal. Geom 41(4):447~460
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