Biharmonic map

In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A biharmonic submanifold refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by James Eells and Luc Lemaire in 1983.Template:Sfnm The study of harmonic maps, of which the study of biharmonic maps is an outgrowth (any harmonic map is also a biharmonic map), had been (and remains) an active field of study for the previous twenty years.Template:Sfnm A simple case of biharmonic maps is given by biharmonic functions.

Definition

Given Riemannian or pseudo-Riemannian manifolds Template:Math and Template:Math, a map Template:Mvar from Template:Mvar to Template:Mvar which is differentiable at least four times is called a biharmonic map if $$\mathrm{\Delta }\mathrm{\Delta }f+\sum _{i=1}^m{R}^h(\mathrm{\Delta }f,df(e_i),df(e_i))=0;$$ given any point Template:Mvar of Template:Mvar, each side of this equation is an element of the tangent space to Template:Mvar at Template:Math.Template:Sfnm In other words, the above equation is an equality of sections of the vector bundle Template:Math. In the equation, Template:Math is an arbitrary Template:Mvar-orthonormal basis of the tangent space to Template:Mvar and Template:Math is the Riemann curvature tensor, following the convention Template:Math. The quantity Template:Math is the "tension field" or "Laplacian" of Template:Mvar, as was introduced by Eells and Sampson in the study of harmonic maps.Template:Sfnm

In terms of the trace, interior product, and pullback operations, the biharmonic map equation can be written as $$\mathrm{\Delta }\mathrm{\Delta }f+{\mathrm{tr}}_g\left(f^{\ast }\right({\iota }_{\mathrm{\Delta }f}{R}^h\left)\right)=0.$$ In terms of local coordinates Template:Math for Template:Mvar and local coordinates Template:Math for Template:Mvar, the biharmonic map equation is written as $$g^{ij}(\frac{\partial }{\partial x^i}(\frac{\partial (\mathrm{\Delta }f{)}^{\alpha }}{\partial x^j}+\frac{\partial f^{\beta }}{\partial x^j}{\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }(\mathrm{\Delta }f{)}^{\gamma })-{\mathrm{\Gamma }}_{ij}^k(\frac{\partial (\mathrm{\Delta }f{)}^{\alpha }}{\partial x^k}+\frac{\partial f^{\beta }}{\partial x^k}{\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }(\mathrm{\Delta }f{)}^{\gamma })+\frac{\partial f^{\delta }}{\partial x^i}{\mathrm{\Gamma }}_{\delta ϵ}^{\alpha }(\frac{\partial (\mathrm{\Delta }f{)}^{ϵ}}{\partial x^j}+\frac{\partial f^{\beta }}{\partial x^j}{\mathrm{\Gamma }}_{\beta \gamma }^{ϵ}(\mathrm{\Delta }f{)}^{\gamma }))+g^{ij}{R}_{\beta \gamma \delta }^{\alpha }(\mathrm{\Delta }f{)}^{\beta }\frac{\partial f^{\gamma }}{\partial x^i}\frac{\partial f^{\delta }}{\partial x^j}=0,$$ in which the Einstein summation convention is used with the following definitions of the Christoffel symbols, Riemann curvature tensor, and tension field: $$\begin{array}{cc}{\mathrm{\Gamma }}_{ij}^k& =\frac{1}{2}{g}^{kl}(\frac{\partial g_{jl}}{\partial x^i}+\frac{\partial g_{il}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^l})\\ {\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }& =\frac{1}{2}{h}^{\alpha \delta }(\frac{\partial h_{\gamma \delta }}{\partial y^{\beta }}+\frac{\partial h_{\beta \delta }}{\partial y^{\gamma }}-\frac{\partial h_{\beta \gamma }}{\partial y^{\delta }})\\ R_{\beta \gamma \delta }^{\alpha }& =\frac{\partial {\mathrm{\Gamma }}_{\gamma \delta }^{\alpha }}{\partial y^{\beta }}-\frac{\partial {\mathrm{\Gamma }}_{\beta \delta }^{\alpha }}{\partial y^{\gamma }}+{\mathrm{\Gamma }}_{\beta \rho }^{\alpha }{\mathrm{\Gamma }}_{\gamma \delta }^{\rho }-{\mathrm{\Gamma }}_{\gamma \rho }^{\alpha }{\mathrm{\Gamma }}_{\beta \delta }^{\rho }\\ {\left(\mathrm{\Delta }f\right)}^{\alpha }& =g^{ij}(\frac{{\partial }^2{f}^{\alpha }}{\partial x^i\partial x^j}-{\mathrm{\Gamma }}_{ij}^k\frac{\partial f^{\alpha }}{\partial x^k}+\frac{\partial f^{\beta }}{\partial x^i}{\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }\frac{\partial f^{\gamma }}{\partial x^j}).\end{array}$$ It is clear from any of these presentations of the equation that any harmonic map is automatically biharmonic. For this reason, a proper biharmonic map refers to a biharmonic map which is not harmonic.

In the special setting where Template:Mvar is a (pseudo-)Riemannian immersion, meaning that it is an immersion and that Template:Mvar is equal to the induced metric Template:Math, one says that one has a biharmonic submanifold instead of a biharmonic map. Since the mean curvature vector of Template:Mvar is equal to the laplacian of Template:Math, one knows that an immersion is minimal if and only if it is harmonic. In particular, any minimal immersion is automatically a biharmonic submanifold. A proper biharmonic submanifold refers to a biharmonic submanifold which is not minimal.

The motivation for the biharmonic map equation is from the bienergy functional $$E_2(f)=\frac{1}{2}\underset{M}{\int }{\left|\mathrm{\Delta }f\right|}_h^{2}d{v}_g,$$ in the setting where Template:Mvar is closed and Template:Mvar and Template:Mvar are both Riemannian; Template:Math denotes the volume measure on $M$ induced by Template:Mvar. Eells & Lemaire, in 1983, suggested the study of critical points of this functional.Template:Sfnm Guo Ying Jiang, in 1986, calculated its first variation formula, thereby finding the above biharmonic map equation as the corresponding Euler-Lagrange equation.Template:Sfnm Harmonic maps correspond to critical points for which the bioenergy functional takes on its minimal possible value of zero.

Examples and classification

A number of examples of biharmonic maps, such as inverses of stereographic projections in the special case of four dimensions, and inversions of punctured Euclidean space, are known.Template:Sfnm There are many examples of biharmonic submanifolds, such as (for any Template:Mvar) the generalized Clifford torus $$\{x\in {ℝ}^{n+2}:x_1^2+\cdots +x_{k+1}^2=x_{k+2}^2+\cdots +x_{n+2}^2=\frac{1}{2}\},$$ as a submanifold of the Template:Math-sphere.Template:Sfnm It is minimal if and only if Template:Mvar is even and equal to Template:Math.

The biharmonic curves in three-dimensional space forms can be studied via the Frenet equations. It follows easily that every constant-speed biharmonic curve in a three-dimensional space form of nonpositive curvature must be geodesic.Template:Sfnm Any constant-speed biharmonic curves in the round three-dimensional sphere Template:Math can be viewed as the solution of a certain constant-coefficient fourth-order linear ordinary differential equation for a Template:Math-valued function.Template:Sfnm As such the situation can be completely analyzed, with the result that any such curve is, up to an isometry of the sphere:

• a constant-speed parametrization of the intersection of Template:Math with the two-dimensional linear subspace Template:Math
• a constant-speed parametrization of the intersection of Template:Math with the two-dimensional affine subspace Template:Math, for any choice of Template:Math which is on the circle of radius Template:Math around the origin in Template:Math
• a constant-speed reparametrization of $$t\mapsto (\frac{\cos at}{\sqrt{2}},\frac{\sin at}{\sqrt{2}},\frac{\cos bt}{\sqrt{2}},\frac{\sin bt}{\sqrt{2}})$$ for any Template:Math on the circle of radius Template:Math around the origin in Template:Math.

In particular, every constant-speed biharmonic curve in Template:Math has constant geodesic curvature.

As a consequence of the purely local study of the Gauss-Codazzi equations and the biharmonic map equation, any connected biharmonic surface in Template:Math must have constant mean curvature.Template:Sfnm If it is nonzero (so that the surface is not minimal) then the second fundamental form must have constant length equal to Template:Math, as follows from the biharmonic map equation. Surfaces with such strong geometric conditions can be completely classified, with the result that any connected biharmonic surface in Template:Math must be either locally (up to isometry) part of the hypersphere $$\left\{\right((w,x,y,\frac{1}{\sqrt{2}}):w^2+x^2+y^2=\frac{1}{2}\},$$ or minimal.Template:Sfnm In a similar way, any biharmonic hypersurface of Euclidean space which has constant mean curvature must be minimal.Template:Sfnm

Guo Ying Jiang showed that if Template:Mvar and Template:Mvar are Riemannian, and if Template:Mvar is closed and Template:Mvar has nonpositive sectional curvature, then a map from Template:Math to Template:Math is biharmonic if and only if it is harmonic.Template:Sfnm The proof is to show that, due to the sectional curvature assumption, the Laplacian of Template:Math is nonnegative, at which point the maximum principle applies. This result and proof can be compared to Eells & Sampson's vanishing theorem, which says that if additionally the Ricci curvature of Template:Mvar is nonnegative, then a map from Template:Math to Template:Math is harmonic if and only if it is totally geodesic.Template:Sfnm As a special case of Jiang's result, a closed submanifold of a Riemannian manifold of nonpositive sectional curvature is biharmonic if and only if it is minimal. Partly based on these results, it was conjectured by R. Caddeo, S. Montaldo and C. Oniciuc that every biharmonic submanifold of a Riemannian manifold of nonpositive sectional curvature must be minimal.Template:Sfnm This, however, is now known to be false.Template:Sfnm The special case of submanifolds of Euclidean space is an older conjecture of Bang-Yen Chen.Template:Sfnm Chen's conjecture has been proven in a number of geometrically special cases.Template:Sfnm

See also

• Bi-Yang–Mills equations, non-linear generalization

References

Bibliography

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