Abstract
In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bruzzo, U., Graña Otero, B.: Metrics on semistable and numerically effective Higgs bundles. J. Reine Angew. Math. 612, 59–79 (2007)
Buchdahl, N.P.: Hermitian–Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280(4), 625–648 (1988)
Cardona, S.A.: On vanishing theorems for Higgs bundles. Differential Geom. Appl. 35, 95–102 (2014)
Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50(1), 1–26 (1985)
Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. (2) 60, 140–145 (1954)
Gauduchon, P.: Torsion 1-forms of compact Hermitian manifolds. Math. Ann. 267(4), 495–518 (1984)
Greene, R.E., Wu, H.: Curvature and complex analysis. Bull. Amer. Math. Soc. 77, 1045–1049 (1971)
Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987)
Kobayashi, S.: Curvature and stability of vector bundles. Proc. Japan Acad. Ser. A Math. Sci. 58(4), 158–162 (1982)
Li, J., Yau, S.T.: Hermitian–Yang–Mills Connection on Non-Kähler Manifolds. Mathematical Aspects of String Theory (San Diego, Calif., 1986), Advanced Series in Mathematical Physics, vol. 1, pp. 560–573. World Scientific Publishing, Singapore (1987)
Lübke, M., Teleman, A.: The Kobayashi–Hitchin Correspondence. World Scientific Publishing Co., Inc., River Edge (1995)
Lübke, M., Teleman, A.: The universal Kobayashi–Hitchin correspondence on Hermitian manifolds. Mem. Amer. Math. Soc. 183(863), vi+97 (2006)
Li, J.Y., Zhang, C.J., Zhang, X.: Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var. Partial Differential Equations 56(3), 81 (2017)
Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. (2) 82, 540–567 (1965)
Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Amer. Math. Soc. 1(4), 867–918 (1988)
Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)
Uhlenbeck, K.K., Yau, S.T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39(S1), S257–S293 (1986)
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25(7), 659–670 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors were supported in part by NSF in China, Nos. 11625106, 11571332 and 11721101.
Rights and permissions
About this article
Cite this article
Li, Y., Zhang, C. & Zhang, X. A Liouville theorem on complete non-Kähler manifolds. Ann Glob Anal Geom 55, 623–629 (2019). https://doi.org/10.1007/s10455-018-9643-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-018-9643-z