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.[1] 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.[2] A simple case of biharmonic maps is given by biharmonic functions.

Definition

Given Riemannian or pseudo-Riemannian manifolds (M, g) and (N, h), a map f from M to N which is differentiable at least four times is called a biharmonic map if

${\displaystyle \Delta \Delta f+\sum _{i=1}^{m}R^{h}{\big (}\Delta f,df(e_{i}),df(e_{i}){\big )}=0;}$

given any point p of M, each side of this equation is an element of the tangent space to N at f(p).[3] In other words, the above equation is an equality of sections of the vector bundle f ^*TN → M. In the equation, e₁, ..., e_m is an arbitrary g-orthonormal basis of the tangent space to M and R^h is the Riemann curvature tensor, following the convention R(u, v, w) = ∇_u∇_vw − ∇_v∇_uw − ∇_{[u, v]}w. The quantity ∆f is the "tension field" or "Laplacian" of f, as was introduced by Eells and Sampson in the study of harmonic maps.[4]

In terms of the trace, interior product, and pullback operations, the biharmonic map equation can be written as

${\displaystyle \Delta \Delta f+\operatorname {tr} _{g}{\Big (}f^{\ast }{\big (}\iota _{\Delta f}R^{h}{\big )}{\Big )}=0.}$

In terms of local coordinates xⁱ for M and local coordinates y^α for N, the biharmonic map equation is written as

${\displaystyle g^{ij}\left({\frac {\partial }{\partial x^{i}}}\left({\frac {\partial (\Delta f)^{\alpha }}{\partial x^{j}}}+{\frac {\partial f^{\beta }}{\partial x^{j}}}\Gamma _{\beta \gamma }^{\alpha }(\Delta f)^{\gamma }\right)-\Gamma _{ij}^{k}\left({\frac {\partial (\Delta f)^{\alpha }}{\partial x^{k}}}+{\frac {\partial f^{\beta }}{\partial x^{k}}}\Gamma _{\beta \gamma }^{\alpha }(\Delta f)^{\gamma }\right)+{\frac {\partial f^{\delta }}{\partial x^{i}}}\Gamma _{\delta \epsilon }^{\alpha }\left({\frac {\partial (\Delta f)^{\epsilon }}{\partial x^{j}}}+{\frac {\partial f^{\beta }}{\partial x^{j}}}\Gamma _{\beta \gamma }^{\epsilon }(\Delta f)^{\gamma }\right)\right)+g^{ij}R_{\beta \gamma \delta }^{\alpha }(\Delta f)^{\beta }{\frac {\partial f^{\gamma }}{\partial x^{i}}}{\frac {\partial f^{\delta }}{\partial x^{j}}}=0,}$

in which the Einstein summation convention is used with the following definitions of the Christoffel symbols, Riemann curvature tensor, and tension field:

${\displaystyle {\begin{aligned}\Gamma _{ij}^{k}&={\frac {1}{2}}g^{kl}{\Big (}{\frac {\partial g_{jl}}{\partial x^{i}}}+{\frac {\partial g_{il}}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{l}}}{\Big )}\\\Gamma _{\beta \gamma }^{\alpha }&={\frac {1}{2}}h^{\alpha \delta }{\Big (}{\frac {\partial h_{\gamma \delta }}{\partial y^{\beta }}}+{\frac {\partial h_{\beta \delta }}{\partial y^{\gamma }}}-{\frac {\partial h_{\beta \gamma }}{\partial y^{\delta }}}{\Big )}\\R_{\beta \gamma \delta }^{\alpha }&={\frac {\partial \Gamma _{\gamma \delta }^{\alpha }}{\partial y^{\beta }}}-{\frac {\partial \Gamma _{\beta \delta }^{\alpha }}{\partial y^{\gamma }}}+\Gamma _{\beta \rho }^{\alpha }\Gamma _{\gamma \delta }^{\rho }-\Gamma _{\gamma \rho }^{\alpha }\Gamma _{\beta \delta }^{\rho }\\(\Delta f)^{\alpha }&=g^{ij}{\Big (}{\frac {\partial ^{2}f^{\alpha }}{\partial x^{i}\partial x^{j}}}-\Gamma _{ij}^{k}{\frac {\partial f^{\alpha }}{\partial x^{k}}}+{\frac {\partial f^{\beta }}{\partial x^{i}}}\Gamma _{\beta \gamma }^{\alpha }{\frac {\partial f^{\gamma }}{\partial x^{j}}}{\Big )}.\end{aligned}}}$

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 f is a (pseudo-)Riemannian immersion, meaning that it is an immersion and that g is equal to the induced metric f ^*h, one says that one has a biharmonic submanifold instead of a biharmonic map. Since the mean curvature vector of f is equal to the laplacian of f : (M, f ^*h) → (N, h), 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

${\displaystyle E_{2}(f)={\frac {1}{2}}\,\int _{M}|\Delta f|_{h}^{2}\,dv_{g},}$

in the setting where M is closed and g and h are both Riemannian; dv_g denotes the volume measure on ${\displaystyle M}$ induced by g. Eells & Lemaire, in 1983, suggested the study of critical points of this functional.[5] Guo Ying Jiang, in 1986, calculated its first variation formula, thereby finding the above biharmonic map equation as the corresponding Euler-Lagrange equation.[6] Harmonic maps correspond to critical points for which the bienergy 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.[7] There are many examples of biharmonic submanifolds, such as (for any k) the generalized Clifford torus

${\displaystyle {\Big \{}x\in \mathbb {R} ^{n+2}:x_{1}^{2}+\cdots +x_{k+1}^{2}=x_{k+2}^{2}+\cdots +x_{n+2}^{2}={\frac {1}{2}}{\Big \}},}$

as a submanifold of the (n + 1)-sphere.[8] It is minimal if and only if n is even and equal to 2k.

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.[9] Any constant-speed biharmonic curves in the round three-dimensional sphere S³ can be viewed as the solution of a certain constant-coefficient fourth-order linear ordinary differential equation for a ℝ⁴-valued function.[10] 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 S³ ⊂ ℝ⁴ with the two-dimensional linear subspace ℝ × ℝ × {0} × {0}
• a constant-speed parametrization of the intersection of S³ ⊂ ℝ⁴ with the two-dimensional affine subspace ℝ × ℝ × {d₁} × {d₂}, for any choice of (d₁, d₂) which is on the circle of radius 2^−1/2 around the origin in ℝ²
• a constant-speed reparametrization of

${\displaystyle t\mapsto {\Big (}{\frac {\cos at}{\sqrt {2}}},{\frac {\sin at}{\sqrt {2}}},{\frac {\cos bt}{\sqrt {2}}},{\frac {\sin bt}{\sqrt {2}}}{\Big )}}$

for any (a, b) on the circle of radius 2^1/2 around the origin in ℝ².

In particular, every constant-speed biharmonic curve in S³ 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 S³ must have constant mean curvature.[11] If it is nonzero (so that the surface is not minimal) then the second fundamental form must have constant length equal to 2^1/2, 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 S³ must be either locally (up to isometry) part of the hypersphere

${\displaystyle \left\{{\Big (}(w,x,y,{\frac {1}{\sqrt {2}}}{\Big )}:w^{2}+x^{2}+y^{2}={\frac {1}{2}}\right\},}$

or minimal.[12] In a similar way, any biharmonic hypersurface of Euclidean space which has constant mean curvature must be minimal.[13]

Guo Ying Jiang showed that if g and h are Riemannian, and if M is closed and h has nonpositive sectional curvature, then a map from (M, g) to (N, h) is biharmonic if and only if it is harmonic.[14] The proof is to show that, due to the sectional curvature assumption, the Laplacian of |∆f|² 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 g is nonnegative, then a map from (M, g) to (N, h) is harmonic if and only if it is totally geodesic.[15] 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 that every biharmonic submanifold of a Riemannian manifold of nonpositive sectional curvature must be minimal.[16] This, however, is now known to be false.[17] The special case of submanifolds of Euclidean space is an older conjecture of Bang-Yen Chen.[18] Chen's conjecture has been proven in a number of geometrically special cases.[19]

References

Footnotes

1. Eells & Lemaire 1983, (8.7) and (8.8).
2. Eells & Sampson 1964.
3. Jiang 1986, Definition 5; Chen 2011, eq. (7.64).
4. Eells & Sampson 1964, p. 116.
5. Eells & Lemaire 1983, (8.7).
6. Jiang 1986, Theorem 3.
7. Montaldo & Oniciuc 2006, Sections 5−7.
8. Jiang 1986, Example 12.
9. Caddeo, Montaldo & Oniciuc 2001, Proposition 3.1.
10. Caddeo, Montaldo & Oniciuc 2001, Proposition 3.2.
11. Caddeo, Montaldo & Oniciuc 2001, Theorem 4.5.
12. Caddeo, Montaldo & Oniciuc 2001, Theorem 4.8.
13. Chen 2011, Corollary 2.10.
14. Jiang 1986, Proposition 7.
15. Eells & Sampson 1964, p. 124.
16. Caddeo, Montaldo & Oniciuc 2001, p. 869.
17. Chen 2011, p. 147.
18. Chen 1991, Conjecture 3; Chen 1996, Conjecture 25.B.6.
19. Chen 1996, Theorems 15.4, 15.6−15.8, 15.10, 15.12−15.13.

Books and surveys

• Bang-Yen Chen. Pseudo-Riemannian geometry, δ-invariants and applications. With a foreword by Leopold Verstraelen. World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2011. xxxii+477 pp. ISBN 978-981-4329-63-7, 981-4329-63-0
• Bang-Yen Chen. Total mean curvature and submanifolds of finite type. Second edition. With a foreword by Leopold Verstraelen. Series in Pure Mathematics, 27. World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2015. xviii+467 pp. ISBN 978-981-4616-69-0, 978-981-4616-68-3
• James Eells and Luc Lemaire. Selected topics in harmonic maps. CBMS Regional Conference Series in Mathematics, 50. Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R.I., 1983. v+85 pp. ISBN 0-8218-0700-5

Articles

• R. Caddeo, S. Montaldo, and C. Oniciuc. "Biharmonic submanifolds of S³". International Journal of Mathematics 12 (2001), no. 8, 867–876. doi:10.1142/S0129167X01001027
• Bang-Yen Chen. "Some open problems and conjectures on submanifolds of finite type". Soochow Journal of Mathematics 17 (1991), no. 2, 169–188.
• Bang-Yen Chen. "A report on submanifolds of finite type". Soochow Journal of Mathematics 22 (1996), no. 2, 117–337.
• James Eells, Jr. and J.H. Sampson. "Harmonic mappings of Riemannian manifolds". American Journal of Mathematics 86 (1964), 109–160. doi:10.2307/2373037
• Guo Ying Jiang. "2-harmonic maps and their first and second variational formulas". Chinese Annals of Mathematics, Series A 7 (1986), no. 4, 389–402. English summary in: Chinese Annals of Mathematics, Series B 7 (1986), no. 4, 523. Translated to English by Hajime Urakawa in: Note di Matematica 28 (2009), suppl. 1, 209–232. doi:10.1285/i15900932v28n1supplp209
• S. Montaldo and C. Oniciuc. "A short survey on biharmonic maps between Riemannian manifolds". Revista de la Unión Matemática Argentina 47 (2006), no. 2, 1–22.