Abstract
Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from S3 embedded in R4. The presence of the cut introduces singular perturbations of the domain, and we develop an analytical framework to characterize well-posed problems in this setting. Our approach combines Green’s functions, spectral analysis, and Sobolev space methods to establish solvability criteria and uniqueness results. In particular, we identify explicit conditions for the existence of solutions with data supported near the cut, and extend the formulation to include delta-type perturbations supported on the removed circle. These results generalize earlier work on punctured two-dimensional spheres and provide a foundation for the study of PDEs on manifolds with localized singularities.