Biharmonic Identity Maps Between Euclidean Spaces and Warped Product Spaces

In this paper, we study biharmonic identity maps between Euclidean spaces and warped product spaces of the form ${\mathbb{R}}^r{\times }_{f^2}{\mathbb{R}}^{n-r}$. Our main results characterize both orientations of the identity map in terms of partial differential equations: For the map from Euclidean space to the warped space, biharmonicity is equivalent to the warping function satisfying a stationary Hamilton–Jacobi-type equation. While the only global solution is constant, we construct infinitely many explicit local solutions. Conversely, for the map from the warped space to Euclidean space, biharmonicity corresponds to a logarithmic transformation of the warping function satisfying this same PDE. This equation admits abundant explicit nonconstant global solutions and can be reduced to a Liouville-type equation via a suitable transformation.