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Let $M^{n}, 2\leq n\leq6$ be a complete noncompact hypersurface immersed in $\HH^{n+1}$. We show that there exist two certain positive constants $0<\delta\leq1$, and $\beta$ depending only on $\delta$ and the first eigenvalue $\lambda_1(M)$ of Laplacian such that if $M$ satisfies a ($\delta$-SC) condition and $\lambda_1(M)$ has a lower bound then $H^{1}(L^{2}(M))=0$. Excepting these two conditions, there is no more additional condition on the curvature.
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참고문헌 (14)
참고문헌 신청
H. D. Cao / 1997 / The structure of stable minimal hypersurfaces in Rn+1 / Math. Res. Lett 4(5):637~644
G. Carron / L2 harmonic forms on non compact manifolds, Lectures given at CIMPA’s summer school “Recent Topics in Geometric analysis” / Arxiv:0704.3194v1
G. Carron / 1998 / Une suite exacte en L2-cohomologie / Duke Math. J 95(2):343~372
L. F. Cheung / 2001 / Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space / Math. Z 236(3):525~530
N. T. Dung / 2015 / Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold / J. Math. Anal. Appl 423(2):1594~1609
G. Carron / L2 harmonic forms on non compact manifolds, Lectures given at CIMPA’s summer school “Recent Topics in Geometric analysis” / Arxiv:0704.3194v1
G. Carron / 1998 / Une suite exacte en L2-cohomologie / Duke Math. J 95(2):343~372
L. F. Cheung / 2001 / Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space / Math. Z 236(3):525~530
N. T. Dung / 2015 / Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold / J. Math. Anal. Appl 423(2):1594~1609
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