Asymptotics of the graph Laplace operator near an isolated singularity

In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $\kappa$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $\kappa$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results.