Abstract
For a Kähler manifold endowed with a weighted measure \(e^{-f}\,dv,\) the associated weighted Hodge Laplacian \(\Delta _{f}\) maps the space of \((p,q)\)-forms to itself if and only if the \((1,0)\)-part of the gradient vector field \(\nabla f\) is holomorphic. We use this fact to prove that for such \(f\), a finite energy \(f\)-harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for \(f\)-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such \(f\)-harmonic maps must be constant if \(f\) has an isolated minimum point. In particular, this implies that for a compact Kähler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calabi, E.: Extremal Kähler metrics. Seminar on Diffential Geometry. Ann. Math. Stud., vol. 102, pp. 259–290. Princeton University Press, Princeton, NJ (1982)
Cao, H.D.: Existence of Gradient Kähler-Ricci Solitons, Elliptic and Parabolic Methods in Geometry. AK Peters, Wellesley (1996)
Cao, H.D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 364, 2377–2391 (2012)
Cao, H.D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Differ. Geom. 85(2), 175–186 (2010)
Carrell, J.B., Lieberman, D.: Holomorphic vector fields and Kähler manifolds. Invent. Math. 21, 275–286 (1973)
Frankel, T.: Fixed points and torsion on Kähler manifolds. Ann. Math. 70, 1–8 (1959)
Gross, L.: Hypercontractivity over complex manifolds. Acta Math. 182, 159–206 (1999)
Gross, L., Qian, Z.: Holomorphic Dirichlet forms on complex manifolds. Math. Z. 246, 521–561 (2004)
Hall, S., Murphy, T.: On the linear stability of Kähler-Ricci solitons. Proc. Am. Math. Soc. 139(9), 3327–3337 (2011)
Howard, A.: Holomorphic vector fields on algebraic manifolds. Am. J. Math. 94, 1282–1290 (1972)
Hua, B., Liu, S., Xia, C.: Liouville theorems for f-harmonic maps into Hadamard spaces. arXiv:1305.0485
Li, P.: On the structure of complete Kähler manifolds with nonnegative curvature near infinity. Invent. Math. 99, 579–600 (1990)
Munteanu, O., Wang, J.: Holomorphic functions on Kähler Ricci solitons. J. Lond. Math. Soc. 89(3), 817–831 (2014)
Munteanu, O., Wang, J.: Topology of Kähler Ricci solitons. To appear in J. Differ. Geom. arXiv:1312.1318
Rimoldi, M., Veronelli, G.: Topology of steady and expanding gradient Ricci solitons via f-harmonic maps. Differ. Geom. Appl. 31, 623–638 (2013)
Schoen, R., Yau, S.T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51(3), 333–341 (1976)
Schoen, R., Yau, S.T.: Compact group actions and the topology of manifolds with non-positive curvature. Topology 18, 361–380 (1979)
Siu, Y.T.: The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math. 112(1), 73–111 (1980)
Udagawa, S.: Compact Kähler manifolds and the eigenvalues of the Laplacian. Colloquium Math. 56(2), 341–349 (1988)
Urakawa, H.: Stability of harmonic maps and eigenvalues of the Laplacian. Trans. Am. Math. Soc. 301(2), 557–589 (1987)
Wright, E.: Killing vector fields and harmonic forms. Trans. Am. Math. Soc. 199, 199–202 (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is partially supported by NSF Grant No. DMS-1262140 and the second author by NSF Grant No. DMS-1105799.
