Biharmonic Maps Research Articles

F-Biharmonic Submanifolds in Space Forms and f-Biharmonic Riemannian Submersions from 3-Manifolds

f-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of f-biharmonic curves in a space form. We also obtain a complete classification of proper f-biharmonic isometric immersions of a developable surface in R3 by proving that a proper f-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into R3 exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as a dual notion of isometric immersions (i.e., submanifolds). We also study f-biharmonicity of Riemannian submersions from 3-manifolds by using the integrability data. Examples are given of proper f-biharmonic Riemannian submersions and f-biharmonic surfaces and curves.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

Doubling annulus Pohožaev type identity and applications to approximate biharmonic maps

Doubling annulus Pohožaev type identity and applications to approximate biharmonic maps

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

On harmonic and biharmonic maps from gradient Ricci solitons

AbstractWe study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two‐dimensional cigar soliton must be harmonic.

Existence of extrinsic polyharmonic maps in critical dimensions

Given two closed Riemannian manifolds M of dimension 2m and N⊂RK, we study the existence problem of extrinsic m-polyharmonic maps in a fixed free homotopy class from M to N. We prove that any energy-minimizing sequence in a fixed free homotopy class subsequently converges locally in Wm,2 to a smooth extrinsic m-polyharmonic map except possibly at most finitely many energy concentration points. Moreover, at an energy concentration point we show that there exists a non-constant smooth extrinsic m-polyharmonic map from R2m to N by blow-up analysis. As a consequence, when the homotopy group π2m(N) is trivial, we prove that there always exists a minimizing extrinsic m-polyharmonic map in every free homotopy class in [M2m,N]. This generalizes the celebrated existence results for harmonic maps (m=1) and biharmonic maps (m=2). The main technical ingredient is an ϵ-regularity for an energy-minimizing sequence, which is new for m-polyharmonic maps and should be of independent interest.

Biharmonic Maps on f-Kenmotsu Manifolds with the Schouten–van Kampen Connection

The object of the present paper was to study biharmonic maps on f-Kenmotsu manifolds and f-Kenmotsu manifolds with the Schouten–van Kampen connection. With the help of this connection, our results provided important insights related to harmonic and biharmonic maps.

Proper biharmonic maps and (2,1)-harmonic morphisms from some wild geometries

In this work we construct a variety of new complex-valued proper biharmonic maps and (2, 1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved.

Some Classifications of Conformal Biharmonic and k-polyharmonic Maps

We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.

Characterization of Biharmonic Hypersurface

The main purpose of this paper is to study biharmonic hypersurface in a quasi-paraSasakian manifold $\mathbb{Q}^{2m+1}$. Biharmonic hypersurfaces are special cases of biharmonic maps and biharmonic maps are the critical points of the bienergy functional. The condition of biharmonicity for non-degenerate hypersurfaces in $\mathbb{Q}^{2m+1}$ is investigated for both cases: either the characteristic vector field of $\mathbb{Q}^{2m+1}$ is the unit normal vector field to the hypersurface or it belongs to the tangent space of the hypersurface. Some relevant examples are also illustrated.

Proper biharmonic maps on tangent bundle

This paper, we define the Mus-Gradient metric on tangent bundle $TM$ by a deformation non-conform of Sasaki metric over an n-dimensional Riemannian manifold $(M, g)$. First we investigate the geometry of the Mus-Gradient metric and we characterize a new class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when all of the factors are Euclidean spaces.

$$L^p$$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

We establish an optimal $$L^p$$ -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot \nabla u) +(\Delta \Omega +G)\cdot \nabla u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is antisymmetric and $$f\in L^p(B^n)$$ , and $$D, E, \Omega , G$$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $$\nabla u$$ and $$\nabla ^2 u$$ . This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm–Rivière, Struwe, and Wang. In particular, our results improve Struwe’s Hölder regularity theorem to any Hölder exponent $$\alpha \in (0,1)$$ when $$f\equiv 0$$ , and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the $$L^p$$ -regularity theory of approximate harmonic maps by Moser to Rivière-Struwe’s second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions $$n\ge 2$$ when $$p>\frac{n}{2}$$ , which partially confirms an interesting expectation by Sharp.

On the $D$-isometric deformation of Kenmotsu manifolds and biharmonic maps

On the $D$-isometric deformation of Kenmotsu manifolds and biharmonic maps

Biharmonic maps and biharmonic submanifolds with small curvature integral

In this article, we study biharmonic maps and biharmonic submanifolds with small curvature integral. Let φ:(Mn,g)→(Nm,h) be a biharmonic map from a complete noncompact Riemannian manifold (Mn,g) into a Riemannian manifold (Nm,h) satisfying that the L2-norm of the tension field of the map is finite. If the domain manifold of the map satisfies a Sobolev inequality and the Ln2-norm of the sectional curvature on the image φ(M) is sufficiently small, then we are able to prove the harmonicity of the biharmonic map. It turns out that the fundamental tone of M is sufficiently large, then such a biharmonic map φ must be harmonic. In case where the map is an isometric immersion, we prove that if M satisfies a Sobolev inequality, then M must be minimal under the assumption that the Ln2-norm of the Ricci curvature on M is sufficiently small. Moreover it is shown that if the fundamental tone of a biharmonic submanifold is sufficiently big, then it is minimal.

Vector fields which are biharmonic maps

Vector fields which are biharmonic maps

On the Biharmonicity of Vector Fields and Unit Vector Fields

Let (M, g) be a compact Riemannian manifold. Equipping its tangent bundle TM (resp. unit tangent bundle \(T_1M\)) with a pseudo-Riemannian g-natural metric G (resp. \({\tilde{G}}\)), we study the biharmonicity of vector fields (resp. unit vector fields) as maps \((M,g) \rightarrow (TM,G)\) (resp. \((M,g) \rightarrow (T_1M,{\tilde{G}})\)), as well as critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) of vector fields (resp. unit tangent bundles) on M. Contrary to the Sasaki metric on TM, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of g-natural metrics on TM, the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to \({\mathfrak {X}}(M)\), but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) without being harmonic.

Unique continuation properties for polyharmonic maps between Riemannian manifolds

AbstractPolyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

Biharmonic Maps on Doubly Warped Product Manifolds

Biharmonic Maps on Doubly Warped Product Manifolds

Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

Abstract In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B 1 ⊂ ℝ 4 {B_{1}\subset\mathbb{R}^{4}} into the sphere 𝕊 n {\mathbb{S}^{n}} . In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into 𝕊 n {\mathbb{S}^{n}} , which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

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].