Asymptotics of the graph Laplace operator near an isolated singularity

TL;DR

This paper studies the asymptotic behavior of the graph Laplace operator near isolated singularities on Riemannian manifolds, showing convergence to the Laplace-Beltrami operator under certain conditions and divergence under others.

Contribution

It provides a detailed analysis of how the graph Laplace operator behaves near singularities, including convergence, divergence, and Taylor expansions, with numerical illustrations.

Findings

Convergence to Laplace-Beltrami operator when curvature doesn't grow too fast.

Divergence of the operator when a conformal factor causes rapid curvature growth.

Explicit Taylor expansion of the operator in specific cases.

Abstract

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.