Biharmonic Maps Research Articles

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension n that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the L^p-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where 2<p<n.

Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book [2] by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two methods of constructions that can be used to produce many examples of generalized harmonic morphisms which are not harmonic morphisms. We also give a complete classification of generalized harmonic morphisms among the projections of a warped product space, which provides infinitely many examples of proper biharmonic Riemannian submersions and conformal submersions from a warped product manifold.

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.

Some results and examples of the biharmonic maps with potential

In this paper, we will study the class of biharmonic maps with potential, in the particular case represented by conformal maps between equidimensional manifolds. Some examples are constructed in particular cases (Euclidean space and sphere).

Biharmonic Maps into S-Space forms

We study in this paper the condition on second fundamental form for biharmonicity of submanifolds in S-space forms.

Equivariant biharmonic maps between manifolds with metrics of signature

We obtain the reduction theorem of equivariant biharmonic maps between warped product manifolds, and apply it to punctured Euclidean spaces, spheres and hyperboloids. We also investigate equivariant biharmonic maps from equivariant manifolds with metrics $$(p, \, q)$$ -signature into equivariant manifolds with $$(p', \, q')$$ -signature, prove the reduction theorem, and provide a few examples.

Some constructions of biharmonic maps on the warped product manifolds

In this paper, we characterize a class of biharmonic maps from and between product manifolds in terms of the warping function. Examples are constructed when one of the factors is either Euclidean space or sphere.

Some results of the [formula omitted]-biharmonic maps and applications

In this paper, we give some properties of the f-biharmonic maps, in the particular case represented by conformal maps between equidimensional manifolds. We construct a new example of the f-biharmonic maps and we characterize the p-biharmonicity of some particular cases.

On the bi-harmonic maps with potential

In this note we characterize the harmonic maps and biharmonic maps with potential, and we prove that every biharmonic map with potential on a complete manifold satisfying some conditions is a harmonic map with potential.

$CR$ rigidity of pseudo harmonic maps and pseudo biharmonic maps

The $CR$ analogue of B.-Y. Chen's conjecture on pseudo biharmonic maps will be shown. Pseudo biharmonic, but not pseudo harmonic, isometric immersions with pseudo parallel pseudo mean curvature vector fields, will be characterized.

LIOUVILLE TYPE THEOREMS FOR TRANSVERSALLY HARMONIC AND BIHARMONIC MAPS

We study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds

Biharmonic holomorphic maps into Kähler manifolds

We study biharmonic holomorphic maps from an almost Hermitian manifold into a Kähler manifold. First, by a simple observation of the curvature term in the biharmonic equation, we establish non-existence results of biharmonic holomorphic maps into Kähler manifolds with non-positive holomorphic bisectional curvature, which extend the similar results of biharmonic maps between Riemannian manifolds. Second, by applying the second variation formula of biharmonic maps, we prove a non-existence result of stable biharmonic holomorphic maps into complex projective space.

Some results on harmonic and bi-harmonic maps

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

Harmonic and biharmonic homomorphisms between Riemannian Lie groups

In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. From this class we develop methods to construct many new examples of biharmonic maps.

Biharmonic maps on kenmotsu manifolds

In this paper we study biharmonic maps on Kenmotsu manifolds. An example for biharmonic map of a three-Kenmotsu manifold is constructed for illustration.

Remarks on the nonexistence of biharmonic maps

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $${\phi : (M, g) \to (N, h)}$$ is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that $${\phi}$$ is a harmonic map if one of the following conditions holds: (i) $${|d\phi|}$$ is bounded in L q (M) and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 \leq q \leq \infty}$$ , $${1 < p < \infty}$$ ; or (ii) $${Vol(M) = \infty}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . In addition, if N has strictly negative sectional curvature, we assume that $${rank\phi(q) \geq 2}$$ for some $${q \in M}$$ and $${\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}$$ , for some $${1 < p < \infty}$$ . These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).

Prediction of NOx Emissions and Fuel Consumption of a City Bus under Real Operating Conditions by Means of Biharmonic Maps

The prediction of pollutant emissions and fuel consumption under real operating conditions of any motor vehicle requires the use of complex mathematical models and experimental tools. In the present research, biharmonic maps (BM) were used to predict NOx (nitrogen oxides) emissions and relative fuel–air ratio (Fr) of a passenger city bus. For the collection of experimental data, an instrumented city bus was tested during real passenger transportation. The data were classified into four dynamic sequences: acceleration, idling, deceleration with fuel consumption, and deceleration without fuel consumption. Among them, the acceleration sequence was selected due to high NOx emissions and high fuel consumption. Experimental results were in good agreement with BM predictions. Significant parameters for predicting NOx concentration were vehicle velocity and relative fuel–air ratio (Fr). While for predicting Fr, significant parameters were the exhaust gas flow (EGF) rate, vehicle velocity, and NOx concentration, BM is a good technique for predicting NOx emissions and Fr of a city bus under transient operation.

Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

Let (M,g) be a four dimensional compact Riemannian manifold with boundary and (N,h) be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from M to N under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from M to N under any Dirichlet boundary condition.

Conformal Maps, Biharmonic Maps, and the Warped Product

In this paper we study some properties of conformal maps between equidimensional manifolds, we construct new example of non-harmonic biharmonic maps and we characterize the biharmonicity of some maps on the warped product manifolds.

Biharmonic and f-biharmonic maps from a 2-sphere

We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order linear ordinary differential equation. As applications, we give a method to produce biharmonic maps and f-biharmonic maps from given biharmonic maps and we construct many examples of biharmonic and f-biharmonic maps from a round sphere $S^2$ and between two round spheres. Our examples include non-conformal proper biharmonic maps $(S^2, f^{-1}g_0)\longrightarrow S^2$ and $(S^2, f^{-1}g_0)\longrightarrow S^n$, or non-conformal f-biharmonic maps $(S^2, g_0)\longrightarrow S^2$ and $(S^2,g_0)\longrightarrow S^n$ from round sphere with two singular points.