Equivariant biharmonic maps between manifolds with metrics of signature
A biharmonic map \(f: (M^n, g) \to (N^m, h)\) from an \(n\)-dimensional Riemanian manifold \(M\) into an \(m\)-dimensional Riemannian manifold \(N\) is a critical point of the bienergy functional given by the integral of square norm of the tension field of \(f\). It is well-known from the variational formula that a harmonic map which is a critical point of the energy functional is automatically biharmonic, but not the converse is true in general. In this paper, the author first obtains the reduction theorem for equivaraint biharmonic maps between warped product manifolds. Let \(\chi (x) : M^n \to N^m\) be a harmonic map with constant energy \(e(\chi)\), and \(\phi(t) : (a, b) \to (c, d)\) be a differentiable function. Consider a map \(f=\phi \cdot \chi : (a, b)\times_{A^2} M \to (c, d)\times_{B^2} N^m\), \(f(t, x) = (\phi(t), \chi(x))\) between two warped product manifolds, where \(A(t)\) and \(B(t)\) are positive differentiable functions defined on \({\mathbb R}\). The author shows that a map \(f = \phi \cdot \chi\) is equivariant biharmonic if and only if the function \(\phi\) satisfies \[ G'' + \frac{n A'(t)}{A(t)}G' - \frac{2e(\chi)}{A^2(t)} \left( B^{'2}(\phi(t)) + B(\phi(t)) B''(\phi(t))\right) G = 0, \] where \[ G = \phi'' + \frac{nA'(t)}{A(t)} - \frac{2e(\chi)}{A^2(t)} B(\phi(t))B'(\phi(t)). \] As corollaries, applying this reduction formula to punctured Euclidean spaces, spheres and hyperboloids, the author obtains reduction formulae for equivariant maps to be biharmonic. Second, the author investigates equivariant biharmonic maps from equivariant manifolds with metrics \((p, q)\)-signature into equivariant manifolds with \((p', q')\)-signature, and proves the similar reduction theorem.
scientific article; zbMATH DE number 6876752
scientific article