F-harmonic Maps Research Articles

A Liouville theorem for F-harmonic maps with moderate divergent energy

In this paper, we obtain a Liouville theorem of F-harmonic maps between complete Riemannian manifolds with moderate divergent F-energy. We assume that F is a concave function and satisfies a differential inequality. We employ Ara's F-stress-energy tensor and the Hessian comparison theorem to prove the main result.

$n$-harmonicity, minimality, conformality and cohomology


 
 
 By studying cohomology classes that are related with n-harmonic morphisms and F-harmonic maps, we augment and extend several results on F-harmonic maps, harmonic maps in [1, 3, 14], p-harmonic morphisms in [17], and also revisit our previous results in [9, 10, 21] on Riemannian submersions and n-harmonic morphisms which are submersions. The results, for example Theorem 3.2 obtaine by utilizing the n-conservation law (2.6), are sharp.
 
 

A Schwarz lemma and a Liouville theorem for generalized harmonic maps

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f-harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V=∇f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V-harmonic maps, as a consequence, any V-harmonic function u, satisfying |u(x)|=o(r(x)), on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow.

On the Generalized of p-harmonic and f-harmonic Maps

On the Generalized of p-harmonic and f-harmonic Maps

Geometry of generalized F-harmonic maps

In this paper, we extend the de nition of F-harmonic maps [1] and, we give the notion of F-biharmonic maps, which is a generalization of biharmonic maps between Riemannian manifolds [3] and ƒ-biharmonic maps [7] and we discuss some conformal properties and the stability of F-harmonic maps. Also, we give a formula to construct some examples of proper F-biharmonic maps. Our results are extensions of [1] and [7].

Some isolation results for f-harmonic maps on weighted Riemannian manifolds with boundary

In this paper, we study f -harmonic (respectively, ( p , f ) -harmonic) maps on a compact weighted Riemannian manifold of nonempty boundary and of positive Bakry–Émery Ricci curvature. We first establish a Bochner–Reilly formula for such maps and deduce therefrom some immediate isolation results. In addition, using a weighted Sobolev inequality obtained in [12] , we prove that, under some energy level depending on the Bakry–Émery curvature of the initial manifold and on the sectional curvature of the final one, the only f -harmonic (respectively, ( p , f ) -harmonic) maps are the constant ones. A gap property of the energy density of f -harmonic maps from a weighted Riemannian manifold to a unit sphere is also obtained.

Liouville theorems for f-harmonic maps into Hadamard spaces

In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.

F-Harmonic Maps between Doubly Warped Product Manifolds

In this paper, some properties of F -harmonic and conformal F -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F -harmonic maps are constructed.

STABLE f-HARMONIC MAPS ON SPHERE

In this paper, we prove that any stable f-harmonic map <TEX>${\psi}$</TEX> from <TEX>${\mathbb{S}}^2$</TEX> to N is a holomorphic or anti-holomorphic map, where N is a <TEX>$K{\ddot{a}}hlerian$</TEX> manifold with non-positive holomorphic bisectional curvature and f is a smooth positive function on the sphere <TEX>${\mathbb{S}}^2$</TEX>with Hess <TEX>$f{\leq}0$</TEX>. We also prove that any stable f-harmonic map <TEX>${\psi}$</TEX> from sphere <TEX>${\mathbb{S}}^n$</TEX> (n > 2) to Riemannian manifold N is constant.

SOME RESULTS OF p-BIHARMONIC MAPS INTO A NON-POSITIVELY CURVED MANIFOLD

Abstract. In this paper, we investigate p-biharmonic maps u : (M,g) →(N,h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifR M |τ(u)| a+p dv g < ∞ andR M |d(u)| 2 dv g < ∞, then u is harmonic, where a ≥ 0 is a nonnegativeconstant and p ≥ 2. We also obtain that any weakly convex p-biharmonichypersurfaces in space formN(c) with c ≤ 0 is minimal. These results giveaffirmative partial answer to Conjecture 2 (generalized Chen’s conjecturefor p-biharmonic submanifolds). 1. IntroductionHarmonic maps play a central roll in geometry. They are critical pointsof the energy E(u) =R M|du| 2 2 dv g for smooth maps between manifolds u :(M,g) → (N,h) and the Euler-Lagrange equation is that tension field τ(u)vanishes. Extensions to the notions of p-harmonic maps, F-harmonic mapsand f-harmonic maps were introduced and many results have been carried out(for instance, see [1, 2, 3, 8, 23]). In 1983, J. Eells and L. Lemaire [10] proposedthe problem to consider the biharmonic maps: they are critical maps of thefunctionalE

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.

Bi-f-harmonic map equations on singly warped product manifolds

Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps. In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensional manifolds and some special maps from or into a warped product manifold.

The nonexistence theorems for F-harmonic maps and F-Yang–Mills fields

Let M be an n(n≥3)-dimensional complete Riemannian manifold with radial curvature K:−a2≤K≤−b2<0 with a≥b>0. In this paper, we consider the F-harmonic maps from M and F-Yang–Mills fields on M. By the monotonicity formulae, we can prove that (1) If (n−2)b≥(2dF−1)a, then every F-harmonic map from M to any Riemannian manifold with finite F-energy is constant, which improves the Dong and Wei's result in [3]; (2) If (n−2)b≥(4dF−1)a, then every F-Yang–Mills fields R∇ with finite F-Yang–Mills energy vanishes on M, which improves the Dong and Wei's result in [3].

F-Harmonic morphisms between Riemannian manifolds

f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known Fuglede-Ishihara characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. We also study the f-harmonicity of conformal immersions.

Liouville Type Theorems of f-Harmonic Maps with Potential

In this paper, we introduce the notion of f-harmonic map with potential H with respect to the functional E f,H . We use the stress-energy tensor to obtain some monotonicity formulas and Liouville type theorems for f-harmonic maps with potential H under some conditions on f and H.

Regularity of Weakly Subelliptic F-Harmonic Maps

Let $\Omega\subseteq\mathbb{R}^{m}$ be a bounded domain, $X=\{X_{1},X_{2},\cdots,X_{k_{0}}\}$ be a H\{o}mander family of vector fields on $\Omega$ whose homogeneous dimension is $Q$. In this paper, we improve the dual inequality obtained by P. Hajlasz and P. Strzelecki in 1998, and take use of it to discuss regularity of weakly subelliptic F-harmonic maps into Riemannian manifolds with transitive isometric transformation groups.

Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

In this paper we give some results on the topology of manifolds with ∞-Bakry–Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yauʼs theory of harmonic maps.

F-Harmonic maps of doubly warped product manifolds

In this paper, we study f-harmonicity of some special maps from or into a doubly warped product manifold. First we recall some properties of doubly twisted product manifolds. After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M × μ,λN can not be proper f-harmonic maps, we use projection maps and product maps to construct nontrivial f-harmonic maps. Thus we obtain some similar results given in [21], such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps. Furthermore, we investigate non-trivial f-harmonicity of the product of two harmonic maps.

Some properties of F-harmonic maps

In this note, we investigate estimates of the Morse index for F-harmonic maps into spheres, our results extend partially those obtained in ([14]) and ([15]) for harmonic and p-harmonic maps.

Kinetical Inflation and Quintessence by F-Harmonic Map

We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic maps. We showed that some F-harmonic map such as the trigonometric functions instead of the scalar field in the lagrangian, allow, in the absence of term of potential, reproduce the inflation. However, there are other F-harmonic maps such as exponential maps which can’t produce the inflation; the pressure and the density of this exponential harmonic field being both of the same sign. On the other hand, these exponential harmonic fields redraw well the phenomenon of the quintessence when the variation of these fields remains weak. The problem of coincidence, however remains.