Weighted Yamabe Heat-Trace Comparison

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq3$, and let $(S^n,g)$ be the round sphere, with Riemannian volume measure ${\,\mathrm{d}} v_g$. Write $\Delta_g=\operatorname{div}_g\nabla_g$. To fix an explicit standard convention for the conformal Laplacian, set \[ Y=-\Delta_g+\frac{n-2}{4(n-1)}\operatorname{Scal}_g. \] The supplied record does not specify the radius of the round sphere or the constant normalization of the conformal Laplacian; these conventions must be used consistently throughout the comparison.

For every $W\in C^\infty(S^n;{\mathbb{R}})$ satisfying \[ W(x)>0\quad\text{for every }x\in S^n, \qquad \int_{S^n}W^{n/2}\,{\,\mathrm{d}} v_g=\operatorname{Vol}_g(S^n), \] define \[ Y_W=W^{-1/2}YW^{-1/2}, \] where $W^{-1/2}$ acts by multiplication. Interpret $Y$ and $Y_W$ as their positive self-adjoint elliptic realizations on $L^2(S^n,{\,\mathrm{d}} v_g)$, with domain $H^2(S^n)$. The sphere is closed, so no boundary condition is imposed. For either operator $A$, write its eigenvalues, counted with multiplicity, as $\lambda_j(A)$, and define its heat trace for $t>0$ by \[ \operatorname{Tr}(e^{-tA})=\sum_{j=1}^{\infty}e^{-t\lambda_j(A)}. \]

The supplied weighted Yamabe heat-trace question asks whether, for every integer $n\geq3$ and every such $W$, \[ \max_{t>0}\left[t^{n/2}\operatorname{Tr}(e^{-tY_W})\right] \leq \max_{t>0}\left[t^{n/2}\operatorname{Tr}(e^{-tY})\right]. \] The notation $\max$ is retained from the supplied statement. Attainment at a positive time is not established in the record, and the parameter interval excludes $t=0$. Whether either occurrence should instead be $\sup$ is an unresolved formulation issue; no such replacement is made here.

Short English statement

For every dimension $n\geq3$, the question compares the largest scaled heat trace of a positively weighted conformal Laplacian with that of the round sphere, under the normalization $\int_{S^n}W^{n/2}\,{\,\mathrm{d}} v_g=\operatorname{Vol}_g(S^n)$. The source uses maxima over positive times, but their attainment and a possible supremum interpretation require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001168$, title Weighted Yamabe Heat-Trace Comparison, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2009; no individual proposer is supplied.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. Supplied report provenance and literature source: https://doi.org/10.4171/owr/2009/06. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "half-line critical inequality": https://arxiv.org/abs/math/0611247. The supplied assessment says that the critical half-line inequality was proved, while no proof or counterexample was found for the weighted Yamabe heat-trace conjecture also contained in the extracted record. No formulation of the half-line inequality is supplied here.
• Literature assessment date: 2026-08-21. The recorded status is "open." These are supplied dated curation and literature-triage claims, not independently verified findings; the source itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not an independent verification performed for this draft.