On a Local Radial Rigidity Phenomenon of Elliptic Systems on Riemannian Manifolds

In this paper, we study local radial rigidity phenomena for a broad class of general-order elliptic systems on Riemannian manifolds via a reduction to singular ordinary differential equations of Euler type. These rigidity properties have natural applications to several geometric problems, including prescribed curvature problems of Yamabe and Paneitz type in conformal geometry, higher-order analogues arising from harmonic maps, and Calabi-type problems on Kähler manifolds. The method applies also to higher-order nonlinear Poisson systems and, more generally, to weighted divergence-form elliptic systems, yielding local uniqueness and existence results for solutions with prescribed initial jets at the singularity.