Harmonic manifolds with minimal horospheres

For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial growth of the volume function.     

The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.     

Existence of Harmonic Functions with Finite Energy on Complete Riemannian Manifolds

Let M be a noncompact complete Riemannian manifold. We consider the existence of harmonic functions with |∇u| ∈ L^p(M).     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

A VANISHING THEOREM ON L~2 HARMONIC FORMS

This paper is concerned with the L2 harmonic forms of a complete noncompact Riemannian manifold, i.e. If M has a pole Q, let 0 < p c >,then Hp={0}. If M has a soul, then similar result is obtained.     

Seven Classes of Harmonic Diffeomorphisms

We deduce two necessary and sufficient conditions for a diffeomorphism $f : M \to \overline{M}$ of a Riemannian manifold (M,g) onto a Riemannian manifold $(\overline{M},\bar g)$ to be harmonic. Using the representation theory of groups, we define in an intrinsic way seven classes of such harmonic diffeomorphisms and partly describe the geometry of each class.     

BOCHNER TECHNIQUE IN REAL FINSLER MANIFOLDS

Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.     

A Global Existence Result For The Heat Flow of Harmonic Maps

We consider the heat flow for harmonic maps from R^m to a compact manifold N. When the L^m norm of the gradient of the initial data is small, we prove the existence of a global solution. We prove a similar result for the boundary value problem, when the boundary of the manifold M maps into a point.     

Harmonic functions under quasi-isometry

Given a complete Riemannian manifold (M, g) with nonnegative sectional curvature outside a compact subset. Let h be another Riemannian metric which is uniformly equivalent to g. It was shown that the dimension of the space of bounded harmonic functions on (M, h) is finite and is the same as of that under metric g, and the dimension of the space spanned by nonnegative harmonic functions on (M, h) is also finite and is the same as of that under metric g. Moreover, bases were constructed for both spaces on (M, h) and precise estimates were established on the asymptotic behavior at infinity for those basic functions.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

Harmonic vector fields on Landsberg manifolds

Let (M,F)$(M,F)$ be a compact boundaryless Landsberg manifold. In this work, a necessary and sufficient condition for a vector field on (M,F)$(M,F)$ to be harmonic is obtained. Next, on a compact boundaryless Finsler manifold of zero flag curvature, a necessary and sufficient condition for a vector field to be harmonic is found. Furthermore, the nonexistence of harmonic vector fields on a compact Landsberg manifold is studied and an upper bound for the first de Rham cohomology group is obtained.     

BOUNDARY REGULARITY FOR WEAK HEAT FLOWS

61. IntroductionLet (M, g) be a compact smooth foemannian manifOld of dimension n with C2 boundary0M, and (N, h) be a smooth compact Riemannian manifolds of dimension k. Assume that(N, h) without boundary is isometrically embedded into the Euclidean space (Rm, (., .)).We assume that Sobolev spaceHl (M, N) = {u E Hl (M; R',.)lu(x) E N for a.e.x E M}and for every u E H1 (M; N), define the energy of u,E(u) = / lVuI'dv, (1.1)j. lVuI'dv, (1.1)where in local coordinate 1VuI' = g"pff 3, …     

A curvature identity on a 6-dimensional Riemannian manifold and its applications

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.     

Harmonic maps on Trans-Sasaki manifolds

In this paper we prove that a (?,J)-holomorphic mapf:M→N (i.e.f _*o?=Jof _*) from a Trans-Sasaki manifold to a nearly Kähler manifold is a harmonic map. We also study the stability of a such map whenM is a compact Trans-Sasaki manifold andN is a Kähler manifold.     

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a K?hler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

ON COMPLETE SPACE-LIKE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR

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The hausdorff dimension of the harmonic class on negatively curved surfaces

We study the regularity of the Hausdorff dimension of the harmonic class of a surface M of negative curvature as a function of the riemannian metric. We prove that it is a C^{r− 3} function of the metric in the Banach manifold of C^r riemannian metrics on M. We also prove regularity results for some asymptotic quantities associated to the Brownian motion on M.     

The Isoenergy Inequality for Harmonic Maps from Rotational Symmetric Manifolds

Let u be a harmonic map from a rotational symmetric manifold M and B a unit ball in M, let E(u|B) be the energy of the map u|B and E(u|∂B) the energy of the map u|∂B, then we obtain the relationship which is called the isoenergy inequality between E(u|B) and E(u|∂B):