Complete Serrin analogue in weighted Riemannian manifolds

Establish a full analogue of Serrin’s theorem for weighted Riemannian manifolds (M, g, e^{-f} dV_g): prove that if a bounded domain Ω admits a smooth function u solving the weighted torsion overdetermined problem Δ_f u = −1 in Ω with boundary conditions u = 0 and ∂_ν u = −c on ∂Ω, then Ω must be a metric ball and u must be radially symmetric, and derive the corresponding geometric rigidity conclusions for the ambient weighted manifold.

Background

The paper studies Serrin’s overdetermined boundary value problem in the setting of weighted Riemannian manifolds, where the Laplacian is replaced by the weighted Laplacian Δ_f and curvature by the Bakry–Émery tensor. While classical results characterize balls and yield rigidity in Euclidean or certain Riemannian settings, a comprehensive extension to the weighted manifold setting is not known.

The authors note that only partial results exist and emphasize that obtaining a result fully paralleling Serrin’s theorem—both the symmetry of solutions and the associated rigidity of the underlying geometry—remains unresolved.