Liouville Theorem For Harmonic Maps Research Articles

Liouville Theorem for harmonic maps from Riemannian manifold with compact boundary

Liouville Theorem for harmonic maps from Riemannian manifold with compact boundary

THE LIOUVILLE PROPERTY FOR PSEUDOHARMONIC MAPS WITH FINITE DIRICHLET ENERGY

In this paper, we first derive the CR Bochner formula and the CR Kato's inequality for pseudoharmonic maps. Secondly, by applying the CR Bochner formula and the CR Kato's inequality we are able to prove the Liouville property for pseudoharmonic maps with finite Dirichlet energy in a complete $(2n+1)$-pseudohermitian manifold. This is served as CR analogue to the Liouville theorem for harmonic maps in Riemannian Geometry.

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.

Complete submanifolds of manifolds of negative curvature

In this article, we give sufficient conditions for a complete non-compact submanifold in a simply connected manifold of negative sectional curvature to have only one end. We also prove some Liouville theorems for harmonic maps from the same kind of submanifolds to manifolds of non-positive curvature.

Complete submanifolds in manifolds of partially non-negative curvature

We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.

Harmonic maps and the topology of manifolds with positive spectrum and stable minimal hypersurfaces

In this paper, we prove two Liouville theorems for harmonic maps and apply them to study the topology of manifolds with positive spectrum and stable minimal hypersurfaces in Riemannian manifolds with non-negative bi-Ricci curvature.

Parabolicity, the divergence theorem for $\delta$-subharmonic functions and applications to the Liouville theorems for harmonic maps

We show that the parabolicity of a manifold is equivalent to the validity of the 'divergence theorem' for some class of $\delta$-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.

A note on divergence of $L^{p}$-integrals of subharmonic functions and its applications

A non $L^{p}$-integrability condition of non-constant non-negative subharmonic functions on a general complete manifold $(M,g)$ is given in an optimal form. As an application in differential geometry, several topics related to parabolicity of manifolds, the Liouville theorem for harmonic maps and conformal deformation of metrics are shown without any assumption on the Ricci curvature of $(M,g)$.

Liouville theorem for harmonic maps with potential

Liouville theorem for harmonic maps with potential

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.

An energy estimate of an exterior problem and a Liouville theorem for harmonic maps

We prove that the exterior harmonic maps, from Rn\\Ω(n ≧ 3) to a bounded strictly convex geodesic ball of some Riemannian manifolds, have finite conformal invariant energy. A consequence of this estimate is a Liouville theorem which states that harmonic maps between Euclidean space (Rn, g0) and Riemannian manifolds are constant maps provided their image at infinity falls into a bounded strictly convex geodesic ball.

Liouville theorems for harmonic maps

We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R m ,g 0) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at ∞.

Correction note to “martingales on manifolds and harmonic maps”

Statements and proofs of theorems 4 and 6 in Kendall (1988) (the paper referred to above) require correction, which is supplied by this short note. In particular it gives a full and correct proof of the probabilistic approach to a general Liouville theorem for harmonic maps (theorem 6 of Kendall, 1988). The note is concluded by a summary of progress made on the open questions posed in Kendall (1988), two of which have now been resolved

Energy estimates and Liouville theorems for harmonic maps

Energy estimates and Liouville theorems for harmonic maps

On the Liouville theorem for harmonic maps

Suppose M M and N N are complete Riemannian manifolds; M M with Ricci curvature bounded below by − A - A , A ⩾ 0 A \geqslant 0 , N N with sectional curvature bounded above by a positive constant K K . Let u : M → N u:M \to N be a harmonic map such that u ( M ) ⊂ B R ( y 0 ) u(M) \subset {B_R}({y_0}) . If B R ( y 0 ) {B_R}({y_0}) lies inside the cut locus of y 0 {y_0} and R > π / 2 K R > \pi /2\sqrt K , then the energy density e ( u ) e(u) of u u is bounded by a constant depending only on A A , K K and R R . If A = 0 A = 0 , then u u is a constant map.