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.
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, ..., em is an arbitrary g-orthonormal basis of the tangent space to M and Rh 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
in which the Einstein summation convention is used with the following definitions of the Christoffel symbols, Riemann curvature tensor, and tension field:
It is clear from any of these presentations of the equation that any harmonic map is automatically biharmonic.
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.
The motivation for the biharmonic map equation is from the bienergy functional
in the setting where M is closed and g and h are both Riemannian; dvg denotes the volume measure on
induced by g. Eells & Lemaire, in 1983, suggested the study of critical points of this functional.
[6] Harmonic maps correspond to critical points for which the bioenergy functional takes on its minimal possible value of zero.
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
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.
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 21/2, 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
[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 by R. Caddeo, S. Montaldo and C. Oniciuc that every biharmonic submanifold of a Riemannian manifold of nonpositive sectional curvature must be minimal.
[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.