# zbMATH — the first resource for mathematics

Bounded harmonic functions on Riemannian manifolds of nonpositive curvature. (English) Zbl 1262.53034
Certain general conditions (I–IV) are imposed on a complete simply-connected Riemannian manifold $$M$$ of nonpositive curvature, which guarantee that $$M$$ supports nontrivial bounded harmonic functions.
Very roughly speaking the conditions (I–IV) are as follows. The sectional curvature of $$M$$ is pinched between a negative constant and zero, the Ricci curvature of $$M$$ is less than a negative constant. Removing a certain point $$O\in M$$, there exists a topological fibration $$M \setminus \{ O\} = \mathcal{A} \times \mathcal {B}$$, such that $$\mathcal{N}=\mathcal{A} \times [0, \infty)$$ is a complete submanifold of $$M$$ and $$O = (a^O, 0)$$ for some $$a^O \in \mathcal{A}$$. Some geometrical properties of the projection of $$M \setminus \{O\}$$ to $$\mathcal{N}$$ (i.e. suitable relations between the angle and its segments, for geodesic triangles) and some pinching on the sectional curvature of $$\mathcal{N}$$ are required.
The main result provides new families of bounded harmonic functions on $$M$$. In particular, it includes the Cartan–Hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains $$\operatorname{Re}_I(n,n)$$ and $$\operatorname{Re}_{II}(n)\;(n\geqslant 2)$$ as special cases.
##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text:
##### References:
 [1] Anterson M.T.: The Dirichlet problem at infinity for manifolds of negative curvature. J. Diff. Geom. 18, 701–721 (1983) [2] Anterson M.T., Schoen R.: Positive harmonic functions on complete manifolds of negative curvature. Ann. Math. 121, 429–461 (1985) · Zbl 0587.53045 · doi:10.2307/1971181 [3] Cartan E.: Sur certaines formesriemanniennes remarquables des geometries á groupe fondamental simple. Ann. Sci. École Norm. Sup. 44, 345–367 (1927) · JFM 53.0393.01 [4] Cartan E.: Sur les domaines bornés homogenes de l’espace de n variables complexes. Abh. Math. Sem. Univ. Hamburg 11, 116–162 (1935) · Zbl 0011.12302 · doi:10.1007/BF02940719 [5] Cartan E.: Sur les fonctions de deux variables complexes et problem de la representative analytique. J. Math. Pure et Appl. 10, 1–114 (1931) · JFM 57.0387.01 [6] Chen Q.: Stability and constant boundary-value problem of harmonic maps with potential. J. Aust. Math. Soc. Ser. A 68, 145–152 (2000) · Zbl 0956.58010 · doi:10.1017/S1446788700001907 [7] Ding Q.: A new Laplacian comparison theorem and the estimate of eigenvalues. Chin. Ann. Math. Ser. B 15, 35–42 (1994) · Zbl 0798.53048 [8] Ding Q., Zhou D.T.: The existence of bounded harmonic functions on Cartan–Hadamard manifolds. Bull. Aust. Math. Soc. 53, 197–207 (1996) · Zbl 0885.53036 · doi:10.1017/S0004972700016919 [9] Hebisch W., Zegarlinski B.: Coercive inequalities on metric measure spaces. J. Funct. Anal. 258, 814–851 (2010) · Zbl 1189.26032 · doi:10.1016/j.jfa.2009.05.016 [10] Hua L.K.: On the theory of automorphic functions of a matrix variable I: Geometrical basis. Am. J. Math. 66, 470–488 (1944) · Zbl 0063.02919 · doi:10.2307/2371910 [11] Hua, L.K.: Harmonic analysis of functions of several comples variable in the classical domains. Transl. Math. Monographs, vol. 6. American Mathematical Society, Providence (1963) [12] Hua L.K., Lu Q.K.: Theory of harmonic functions in classical domains. Scientia Sinica 8, 1031–1094 (1959) · Zbl 0090.29503 [13] Lu Q.K.: The classcial manifolds and calssical domains (in Chinese). Shanghai Press of Science and Technology, Shanghai (1963) [14] Noda T., Oda M.: Laplacian comparison and sub-mean-value theorem for multiplier Hermitian manifolds . J. Math. Soc. Japan 56, 1211–1219 (2004) · Zbl 1064.32017 · doi:10.2969/jmsj/1190905456 [15] Schoen R., Yau S.T.: Lectures on Differential Geometry. Internaional Press, Boston (1994) · Zbl 0830.53001 [16] Siegel C.L.: Symplectic geometry. Am. J. Math. 55, 1–86 (1943) · Zbl 0138.31401 · doi:10.2307/2371774 [17] Sullivan D.: The Dirichlet problem at infinity for a negative curved manifold. J. Diff. Geom. 18, 723–732 (1983) · Zbl 0541.53037 [18] Wang F.Y.: Functional inequalities, Markov semigroup and spectral theory. Science Press, Beijing/New York (2005) [19] Weil A.: L’intégration dans les groups topoloiques et ses applications. Hermann, Paris (1940) [20] Weyl H.: Harmonics on homogenous manifolds. Ann. Math. 35, 486–499 (1934) · Zbl 0010.01201 · doi:10.2307/1968746 [21] Wong Y.C.: Euclidean n-planes in pseudo-Euclidean spaces and differential geometry of Cartan domains. Bull. AMS 75, 409–414 (1969) · Zbl 0184.25403 · doi:10.1090/S0002-9904-1969-12198-1 [22] Yau S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975) · Zbl 0297.31005 · doi:10.1002/cpa.3160280203 [23] Yau S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051 [24] Yau, S.T.: Open problems in geometry. In: Greens, R., Yau, S.T. (eds.) Differential Geometry: Partial differential equations on manifolds (Los Angeles, CA 1990) 1–28, Proc. Symps. Pure Math., vol. 54, Part I. AMS, Providence (1993) [25] Xin Y.L.: Harmonic maps of bounded symmetric domains. Math. Ann. 303, 417–433 (1995) · Zbl 0842.58015 · doi:10.1007/BF01460998
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.