Abstract
This paper contains some vanishing theorems for \(L^{2}\) harmonic forms on complete Riemannian manifolds with a weighted Poincaré inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.
Similar content being viewed by others
References
Calderbank, D.M., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173, 214–255 (2000)
Candel, A.: Eigenvalue estimates for minimal surfaces in hyperbolic space. Trans. Am. Math. Soc. 359, 3567–3575 (2007)
Cao, H.-D., Shen, Y., Zhu, S.: The structure of stable minimal hypersurfaces in \(R^{n+1}\). Math. Res. Lett. 4, 637–644 (1997)
Cheng, X.: \(L^{2}\) harmonic forms and stability of hypersurfaces with constant mean curvature. Bol. Soc. Brasil. Mat. (N.S.) 31(2), 225–239 (2000)
Cheng, X.: One end theorem and application to stable minimal hypersurfaces. Arch. Math. 90, 461–470 (2008)
Cheng, X., Cheung, L.-F., Zhou, D.: The structure of weakly stable constant mean curvature hypersurfaces. Tohoku Math. J. 60, 101–121 (2008)
Cheung, L.-F., Leung, P.-F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236, 525–530 (2001)
Cibotaru, D., Zhu, P.: Refined Kato inequalities for harmonic fields on Kähler manifolds. Pac. J. Math. 256, 51–66 (2012)
Dung, N.T., Seo, K.: Vanishing theorems for \(L^{2}\) harmonic 1-forms on complete submanifolds in a Riemannian manifold. J. Math. Anal. Appl. 423(2), 1594–1609 (2015)
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math. 33, 199–211 (1980)
Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)
Kim, J.-J., Yun, G.: On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and \(L^{2}\) harmonic forms. Arch. Math. 100(4), 369–380 (2013)
Kong, S., Li, P., Zhou, D.: Spectrum of the Laplacian on quaternionic Kähler manifolds. J. Differ. Geom. 78, 295–332 (2008)
Lam, K.-H.: Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc. 362, 5043–5062 (2010)
Li, P.: Geometric analysis. In: Cambridge studies in advanced mathematics, vol. 134, pp. x+406. Cambridge University Press, Cambridge (2012)
Li, P., Tam, L.-F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 35, 359–383 (1992)
Li, P., Wang, J.: Complete manifolds with positive spectrum. J. Differ. Geom. 58, 501–534 (2001)
Li, P., Wang, J.: Complete manifolds with positive spectrum, II. J. Differ Geom. 62, 143–162 (2002)
Li, P., Wang, J.: Stable minimal hypersurfaces in a nonnegatively curved manifold. J. Reine Angew. Math. 566, 215–230 (2004)
Li, P., Wang, J.: Comparison theorem for Kähler manifolds and positivity of spectrum. J. Differ. Geom. 69, 43–74 (2005)
Li, P., Wang, J.: Weighted Poincaré inequality and rigidity of complete manifolds. Ann. Sci. École Norm. Sup. 39, 921–982 (2006)
Li, P., Wang, J.: Connectedness at infinity of complete Kähler manifolds. Am. J. Math. 131, 771–817 (2009)
Miyaoka, R.: \(L^{2}\) harmonic 1-forms on a complete stable minimal hypersurface. In: Geometry and Global Analysis (Sendai, 1993), pp. 289–293. Tohoku Univ., Sendai (1993)
Palmer, B.: Stability of minimal hypersurfaces. Comment. Math. Helv. 66, 185–188 (1991)
Seo, K.: Stable minimal hypersurfaces in the hyperbolic space. J. Korean Math. Soc. 48, 253–266 (2011)
Tanno, S.: \(L^{2}\) harmonic forms and stability of minimal hypersurfaces. J. Math. Soc. Jpn. 48, 761–768 (1996)
Shen, Y., Ye, R.: On stable minimal surfaces in manifolds of positive bi-Ricci curvatures. Duke Math. J. 85, 109–116 (1996)
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vieira, M. Vanishing theorems for \(L^{2}\) harmonic forms on complete Riemannian manifolds. Geom Dedicata 184, 175–191 (2016). https://doi.org/10.1007/s10711-016-0165-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-016-0165-1