Can you hear an orbifold singularity?

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Riemannian orbifold is locally a quotient $\widetilde U/G_U$, where $\widetilde U\subset\mathbb R^n$ is open and connected, $G_U$ is a finite group acting effectively by diffeomorphisms, and a $G_U$-invariant Riemannian metric is compatible with the other charts. A point is singular if the stabilizer of a lift is nontrivial. An orbifold with no singular points is a smooth Riemannian manifold. Use the closed, compact and connected setting of the cited restatement. Two such orbifolds are Laplace-isospectral if the eigenvalues of the Laplace operator on smooth functions agree, including multiplicities.

Statement recorded in the supplied source.

Do there exist Laplace-isospectral Riemannian orbifolds $O_1,O_2$ such that $O_1$ has a nonempty singular set and $O_2$ has no singular points? Equivalently, can the Laplace spectrum always detect the presence of an orbifold singularity? [S2]

Short English statement

Can a compact orbifold with singular points and a smooth manifold have exactly the same function-Laplacian spectrum?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6800001 (AMR-067-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Sean Richardson and Elizabeth Stanhope, You Can Hear the Local Orientability of an Orbifold, arXiv:1910.03224v1 (2019), Introduction, printed p. 2; Definition 2.1, pp. 2--3, and Laplace-spectrum convention, p. 4. https://arxiv.org/abs/1910.03224v1.