In this article, we define Perelman’s functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman’s \(\lambda \) -functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator \(-4\Delta + R\) on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the \(\lambda \) -functional on these singular manifolds, and further, to prove that the infimum of W-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the W-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of \(-4\Delta + R\) and Perelman’s functionals on manifolds with more general singularities, i.e. the \(r^{\alpha }\) -horn singularities which serve as prototypes of algebraic singularities.
