Stable Harmonic Maps Research Articles

Dirichlet problem for weakly harmonic maps with rough data

Weakly harmonic maps from a domain (the upper half-space or a bounded domain, ) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes or we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, Moreover, we show that boundary data can be chosen large in the underlying topologies if Ω is smooth and bounded by perturbing strictly stable smooth harmonic maps.

On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces

We construct a family of Hermitian metrics on the Hopf surface S 3 × S 1 \mathbb {S}^3\times \mathbb {S}^1 , whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of π : S 3 × S 1 → C P 1 \pi :\mathbb {S}^3\times \mathbb {S}^1\to \mathbb {C} P^1 two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces. A similar result is true for S 2 n − 1 × S 1 \mathbb {S}^{2n-1}\times \mathbb {S}^{1} .

Semi-slant Riemannian maps into almost Hermitian manifolds

We introduce semi-slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of semi-slant immersions, invariant Riemannian maps, anti-invariant Riemannian maps and slant Riemannian maps. We obtain characterizations, investigate the harmonicity of such maps and find necessary and sufficient conditions for semi-slant Riemannian maps to be totally geodesic. Then we relate the notion of semi-slant Riemannian maps to the notion of pseudo-horizontally weakly conformal maps, which are useful for proving various complex-analytic properties of stable harmonic maps from complex projective space and give many examples of such maps.

SLANT RIEMANNIAN MAPS TO KÄHLER MANIFOLDS

We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficient conditions for slant Riemannian maps to be totally geodesic. Moreover, we relate the notion of slant Riemannian maps to the notion of pseudo horizontally weakly conformal (PHWC) maps which are useful for proving various complex-analytic properties of stable harmonic maps from complex projective space.

STABLE HARMONIC MAPS BETWEEN FINSLER MANIFOLDS AND SSU MANIFOLDS

Using the properties of Cartan tensor, we rewrite the second variation formula for harmonic maps between Finsler manifolds, and we prove that there is no non-degenerate stable harmonic map from a compact SSU manifold to any Finsler manifold, which is obtained by Howard and Wei for the Riemannian case. We also include a proof of a theorem of Shen–Wei which states that there is no non-degenerate stable harmonic map from a compact Finsler manifold to any SSU manifold, by the same second variational formula (see Eq. (2.1) in [Y. B. Shen and S. W. Wei, The stability of harmonic maps on Finster manifolds, Houston J. Math. 34 (2008) 1049–1056]) and the same method [S. W. Wei, An extrinsic average variational method, in Recent Developments in Geometry, Contemporary Mathematics, Vol. 101 (American Mathematical Society, Providence, RI, 1989), pp. 55–78].

Stable harmonic maps between Finsler manifolds and Riemannian manifolds with positive Ricci curvature

We study the stability of harmonic maps between Finsler manifolds and Riemannian manifolds with positive Ricci curvature, and we prove that if M is a compact Einstein Riemannian minimal submanifold of a Riemannian unit sphere with Ricci curvature satisfying Ric > n/2, then there is no non-degenerate stable harmonic map between M and any compact Finsler manifold.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

Second variation of harmonic maps between Finsler manifolds

The first and second variation formulas of the energy functional for a nondegenerate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.

Stability and singularities of harmonic maps into 3-spheres

The behavior of harmonic maps from a domain in R 4 into S 3 is discussed. In this paper, we show that if u is a strictly stable stationary harmonic map : B 4→ S 3 , such that the singular set Sing( u) of u consists of {0}, then the degree deg( u,0) of u at 0 is zero; and that if u is a weakly stable harmonic map B 4→ S 3 with Sing( u)={0}, then deg( u,0)=0 or ±1. Furthermore, the characterization of homogeneous stable stationary harmonic maps from B 4 into S 3 is given.

Stable harmonic maps into the complex projective spaces

Stable harmonic maps into the complex projective spaces

Uniqueness ofx/¦x¦ as a stable configuration in liquid crystals

We deduce the uniqueness ofx/¦x¦ as a stable configuration in liquid crystals for some range of the elastic constants, following an observation on the curvature of the integral curves along the optic axis and a careful study of F.-H. Lin’s proof ofx/¦x¦ as a stable harmonic map. Our study includes the harmonic case when all the elastic constants are equal.

STABLE HARMONIC MAPS FROM COMPLETE MANIFOLDS

Recently, the Liouville theorem for stable harmonic maps is proved from Euclidean space into sphere. The technique employed in this paper allows the researchers to consider stable harmonic maps from complete manifolds.

Non-Existence of Stable Harmonic Maps from Sufficiently Pinched Simply Connected Riemannian Manifolds

Non-Existence of Stable Harmonic Maps from Sufficiently Pinched Simply Connected Riemannian Manifolds

Pinching and nonexistence of stable harmonic maps

Pinching and nonexistence of stable harmonic maps

Stable Harmonic Maps from Riemann Surfaces to Compact Hermitian Symmetric Spaces

In the case where $\Sigma$ is a Riemann sphere, the above result was obtained by Siu [S-l, $Z$] (see also [B-R-S]). In the case where $M$ is a complex projective space and deg $f|\geqq m(p-1)/(m+1)$ where $p$ denotes the genus of $\Sigma$ , the above result was obtained by Eells and Wood [E-W]. They used algebraic geometric arguments (theorems of Riemann-Roch and Grothendieck). Recently, by using a twistor space over the domain manifold, Burns and de Bartolomeis have shown the above result in the case where $M$ is a complex projective space (cf. Remark 6 of [B-R-S]). We show the above theorem by a simple argument for the second variations used in [L-S].

Harmonic Maps and a Pinching Theorem for Positively Curved Hypersurfaces

In this paper, we establish a theorem of Liouville type for stable harmonic maps in sufficiently pinched, positively curved hypersurfaces of a space form with nonnegative constant curvature. Similar results for the Euclidean sphere ${S^n}$ have been proved by Y. L. Xin and P. F. Leung, respectively.

Harmonic maps and a pinching theorem for positively curved hypersurfaces

In this paper, we establish a theorem of Liouville type for stable harmonic maps in sufficiently pinched, positively curved hypersurfaces of a space form with nonnegative constant curvature. Similar results for the Euclidean sphere S n {S^n} have been proved by Y. L. Xin and P. F. Leung, respectively.

Nonexistence of Stable Harmonic Maps To and From Certain Homogeneous Spaces and Submanifolds of Euclidean Space

Call a compact Riemannian manifold $M$ a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from $M$ contains elements of arbitrarily small energy. If $M$ is isometrically immersed in Euclidean space, then a condition on the second fundamental form of $M$ is given which implies $M$ is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.

Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space

Call a compact Riemannian manifold M M a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from M M contains elements of arbitrarily small energy. If M M is isometrically immersed in Euclidean space, then a condition on the second fundamental form of M M is given which implies M M is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.

Stability of harmonic maps and minimal immersions

It was proved by J. Simons [10] that there does not exist any stable minimal submanifold in the Euclidean sphere S n {S^n} , and P. F. Leung proved that any stable harmonic map from any Riemannian manifold to S n {S^n} , where n ⩾ 3 n \geqslant 3 , is a constant. In this paper, we generalize their results and indicate that there are many manifolds having such properties as S n {S^n} .