A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

Liouville theorem for harmonic maps with potential

Let M, N be complete manifolds, u:M→N be a harmonic map with potential H, namely, a critical point of the functional , where e(u) is the energy density of u. We will give a Liouville theorem for u with a class of potentials H's. Received: Received: 10 July 1997     

Liouville theorem for harmonic maps with potential

LetM, N be complete manifolds,u:M →N be a harmonic map with potentialH, namely, a critical point of the functionalE _H (u)=∫ _M [e(u) − H(u)], wheree(u) is the energy density ofu. We will give a Liouville theorem foru with a class of potentialsH’s. Research supported in part by NNSFC, SFECC and NSFCCNU.     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

Liouville theorems for exponentially harmonic functions on Riemannian manifolds

Suppose thatM is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that any bounded exponentially harmonic function onM is a constant function.     

On stable quasi-harmonic maps and Liouville type theorems

We consider Liouville type problems of stable quasi-harmonic maps, by ``stable' we mean that the second variation of quasi-energy functional is nonnegative, and we prove that the stable quasi-harmonic maps must be constant under some geometry conditions.

     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

Liouville theorems for non-local operators

The paper characterizes some classes of pseudo-differential operators for which there are (or there are not) non-constant bounded harmonic functions. Non-local perturbations of Ornstein-Uhlenbeck operators and operators with dissipative coefficients are considered. The methods used are probabilistic and based on the concept of absorption function and on a new extension of the Bismut-Elworthy-Li formula. The probabilistic interpretation of the Liouville theorem by means of absorption functions for general Markov processes is given as well.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Fixed point and rigidity theorems for harmonic maps into NPC spaces

We show that harmonic maps from 2-dimensional Euclidean polyhedra to arbitrary NPC spaces are totally geodesic or constant depending on a geometric and combinatorial condition of the links of the 0-dimensional skeleton. Our method is based on a monotonicity formula rather than a codimension estimate of the singular set as developed by Gromov–Schoen or the mollification technique of Korevaar–Schoen.      

Liouville theorems for III-posed problems

The growth rate at infinity of non-constant solutions of certain abstract differential equations is studied making essential use of the convexity of the norm of the solution.     

Liouville theorems for quasi-harmonic functions

Let N be a compact Riemannian manifold. A self-similar solution for the heat flow is a harmonic map from to N (n≥3), which was also called a quasi-harmonic sphere (cf. Lin and Wang (1999) [1]). (Here is the Euclidean metric in .) It arises from the blow-up analysis of the heat flow at a singular point. When and without the energy constraint, we call this a quasi-harmonic function. In this paper, we prove that there is neither a nonconstant positive quasi-harmonic function nor a nonconstant quasi-harmonic function. However, for all 1≤p≤n/(n−2), there exists a nonconstant quasi-harmonic function in .     

Liouville theorems for some nonlinear inequalities

We prove various Liouville theorems for integral and differential inequalities on the whole ℝR ^N . The main tools we use throughout this paper are representation formulae for linear inequalities, the nonlinear capacity method and the weak form of Harnack’s inequality. This paper is dedicated to Professor Stanislav I. Pohozaev with admiration and friendship     

Extensions of Liouville theorems

The method of deriving Liouville's theorem for subharmonic functions in the plane from the corresponding Hadamard three-circles theorem is extended to a more general and abstract setting. Two extensions of Liouville's theorem for vector-valued holomorphic functions of several complex variables are also mentioned.