Related papers: A Liouville theorem on asymptotically Calabi space…
In this paper we extend Yau's celebrated Liouville theorem to the biharmonic case. Namely, we show that in a complete Riemannian manifold with a pole and nonnegative Ricci curvature, any biharmonic function of subquadratic growth must be…
In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In…
In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient…
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no…
In this manuscript, we investigate the exponentially harmonic equation on noncompact forward complete Finsler metric measure spaces. We demonstrate that this Finslerian equation represents a critical point of an exponential energy…
We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-\'Emery Ricci curvature is nonnegative and its diameter of geodesic…
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…
In this paper, we establish a complete Liouville--type hierarchy for polyharmonic functions on Riemannian manifolds with nonnegative Ricci curvature. Extending Yau's classical result for harmonic functions and our recent biharmonic…
We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…
We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic…
For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite…
In this paper, we study functional and geometric inequalities on complete Finsler measure spaces under the condition that the weighted Ricci curvature ${\rm Ric}_\infty$ has a lower bound. We first obtain some local uniform Poincar\'{e}…
We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete…
We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete K\"{a}hler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of…
On a complete Calabi-Yau manifold $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving…
In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of…
We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive…
In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse…
We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31,…
We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with…