A Parametrization of Solutions to the Yamabe Equation with Isolated Singularity

The Yamabe equation arises naturally in differential geometry, with its solutions corresponding to conformal metrics of constant scalar curvature. In this talk, we focus on solutions defined on a punctured ball in \(\mathbb {R}^n\) that exhibit a nonremovable singularity at the center. I will begin with a brief historical overview of the study of singular Yamabe equations and then present recent progress in this direction. In joint work with Professor Qing Han, we parametrize solutions to the Yamabe equation with nonremovable isolated singularities using harmonic functions, or more precisely, sequences of spherical harmonics. Previous results have shown that such singular solutions can be formally represented by infinite series determined by these sequences. We prove that these formal series indeed converge to genuine solutions of the Yamabe equation with a nonremovable isolated singularity. Moreover, these solutions have designated asymptotic behaviors determined by the sequences of spherical harmonics. We also study the structure of the collection of all such solutions in an appropriate function space. In a broader context, we provide a general method of constructing all solutions near its isolated singular point and all solutions near infinity.