Essential Self-Adjointness on Almost-Riemannian Regular Regions

Definitions and mathematical statement

Let $M$ carry an almost-Riemannian structure. In the usual local description, this structure is represented by $n=\dim M$ smooth vector fields $X_1,\ldots,X_n$, declared orthonormal wherever they are linearly independent. Its regular region is \[ R=\{p\in M:\dim\operatorname{span}\{X_1(p),\ldots,X_n(p)\}=n\}. \] On $R$, the structure induces a Riemannian metric $g$. Write $\mu_g$ for its Riemannian volume measure and $L^2(R,\mu_g)$ for the Hilbert space of square-integrable complex-valued functions. Let $C_c^\infty(R)$ denote the smooth functions with compact support in $R$. With the sign convention $\Delta_g=\operatorname{div}_g\operatorname{grad}_g$, consider the densely defined symmetric operator \[ A:C_c^\infty(R)\subset L^2(R,\mu_g)\longrightarrow L^2(R,\mu_g), \qquad Au=\Delta_g u. \] Essential self-adjointness means that the operator closure $\overline A$ is self-adjoint, equivalently $\overline A=A^*$. Reversing the sign of the Laplacian does not change this property.

Question as supplied. For every almost-Riemannian structure in the intended class, is $A$ essentially self-adjoint? In particular, does this hold for every real-analytic almost-Riemannian structure on a two-dimensional manifold having no tangency points?

For the usual two-dimensional situation in which the singular set $\Sigma=M\setminus R$ is locally a smooth curve and the distribution \[ \mathcal D_p=\operatorname{span}\{X_1(p),X_2(p)\} \] has dimension one at $p\in\Sigma$, a tangency point is a point satisfying $\mathcal D_p=T_p\Sigma$. Thus absence of tangency points means $\mathcal D_p\ne T_p\Sigma$ at every singular point to which this description applies.

Unresolved scope qualifications. The supplied record does not specify the complete definition of the admissible almost-Riemannian structures, including bracket-generation or singular-set regularity hypotheses. The description of tangency above therefore explains the usual smooth-curve case without resolving the treatment of more degenerate singularities. No compactness, completeness, boundary, or control-of-ends assumptions are supplied. These omissions prevent identification of a fully specified universal class and must be checked against the source before treating the question as a precise conjecture. No such hypotheses are added here, and no boundary conditions or self-adjoint extension are selected.

Short English statement

The question asks whether the Laplace--Beltrami operator on the regular region of an almost-Riemannian structure has a unique self-adjoint extension from compactly supported smooth functions. It singles out real-analytic surfaces without tangency points. The precise admissible structures and global hypotheses remain unspecified. The supplied partially solved label and literature assessment are dated curation, not independently verified conclusions.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30004271$, source problem code catalogue entry, title Essential Self-Adjointness on Almost-Riemannian Regular Regions, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Self-adjoint Extensions in New Settings (2020), Oberwolfach Reports, pp. 2911--2950, DOI 10.4171/owr/2019/47. https://doi.org/10.4171/owr/2019/47. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Franceschi--Prandi--Rizzi. https://doi.org/10.1007/s11118-018-09760-w.
• Supplied literature source: OWR 2019 47. https://ems.press/content/serial-article-files/46828.
• Supplied literature source: Quantum confinement on non-complete Riemannian manifolds. https://ems.press/content/serial-article-files/33671?nt=1.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem partially solved and reports essential self-adjointness under important regularity and non-characteristic hypotheses, while describing the broad real-analytic no-tangency conjecture in the extracted record as unresolved for general almost-Riemannian structures. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified, with an explanation that mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This is a supplied editorial qualification, not mathematical verification of this draft or of the conjecture.