Spectral Rigidity of Unit-Area Riemannian Two-Spheres

Definitions and mathematical statement

Let $(M,g)$ be a closed Riemannian surface homeomorphic to $S^2$, with $\operatorname{Area}_g(M)=1$. Write $\lambda_1(g)$ for the first positive eigenvalue of the nonnegative Laplace--Beltrami operator $-\Delta_g$ on functions. The symbol $g_0$ denotes the reference metric on $S^2$ used in the supplied statement; its definition and normalization are not supplied.

For a specified metric $h$ on $S^2$, one standard inhomogeneous Sobolev convention is \[ \|u\|_{H^1(S^2,h)}^2=\int_{S^2}\left(|u|^2+|\nabla_hu|_h^2\right)\,\mathrm dV_h, \qquad H^{-1}(S^2,h)=\left(H^1(S^2,h)\right)^*. \] Under this convention, a smooth function $s$ defines a functional by $u\mapsto\int_{S^2}su\,\mathrm dV_h$, with dual norm \[ \|s\|_{H^{-1}(S^2,h)}=\sup_{0\ne u\in H^1(S^2,h)}\frac{\left|\int_{S^2}su\,\mathrm dV_h\right|}{\|u\|_{H^1(S^2,h)}}. \] These definitions describe a possible convention; the supplied record does not fix the Sobolev norm or pairing convention.

Proposed assertion, retaining the unresolved source notation. Does there exist a universal constant $C>0$ such that, for every unit-area Riemannian topological two-sphere $(M,g)$, there are a diffeomorphism $\Phi:S^2\to M$ and a positive conformal factor $s:S^2\to(0,\infty)$ satisfying \[ \Phi^*g=s g_0, \qquad \|s\|_{H^{-1}(S^2,g)}\le C\lambda_1(g)? \] Here $C$ must be independent of $(M,g)$. The metric identity implies $\int_{S^2}s\,\mathrm dV_{g_0}=1$. The displayed bound contains no additional spectral-defect term.

The notation $H^{-1}(S^2,g)$ needs clarification because $g$ is initially a metric on $M$. Transporting it by $\Phi$, so that the metric in this norm is $\Phi^*g=s g_0$, is a possible reading, but the supplied record does not explicitly settle the intended identification or norm convention. Consequently, the assertion remains a qualified formulation pending clarification of these choices and of $g_0$.

The original statement additionally describes this as asking whether the conformal factors are uniformly bounded in the dual of $H^1$. That claimed equivalence is retained as source wording: a uniform bound and the displayed eigenvalue-dependent estimate are not identified here without further justification.

Short English statement

The supplied question asks whether every unit-area Riemannian two-sphere admits a conformal parametrization whose positive conformal factor has $H^{-1}$ norm at most a universal constant times its first positive Laplace eigenvalue. The reference metric, Sobolev conventions, and claimed equivalence with a uniform dual bound require clarification. The supplied literature assessment concerns a different problem about plane domains.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001159$, exact source problem code catalogue entry, titled "Spectral Rigidity of Unit-Area Riemannian Two-Spheres." The record assigns the problem to 2009 and supplies no individual proposer.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06: https://doi.org/10.4171/owr/2009/06. The record attributes the question to Conjecture 3 and reports a literal check against the official report, with no additional spectral-defect term. It also reports that the source notes conformality of sphere metrics. These are supplied verification claims, not independently verified here.
• Supplied literature source labeled "Henrot, shape optimization and spectral theory": https://doi.org/10.1515/9783110550887.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. The record labels the problem open, but its assessment discusses unresolved local-minimizer regularity and a resulting uniqueness statement for unrestricted metrics on plane domains, and says that smooth perturbations of the disk do not settle that question. This assessment does not match the sphere formulation. The background explicitly calls for independent verification of time-sensitive status and cited claims.