Monotonicity of Weighted Yamabe Heat Traces

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq4$, and let $(S^n,g_{\mathrm{rd}})$ be the unit round sphere, a compact manifold without boundary. Write ${\,\mathrm{d}} v_{\mathrm{rd}}$ for its volume measure and $\operatorname{vol}_{\mathrm{rd}}(S^n)$ for its volume. Use the convention $-\Delta_{g_{\mathrm{rd}}}\geq0$ and the following normalization of the round conformal Laplacian: \[ Y=-\Delta_{g_{\mathrm{rd}}}+\frac{n-2}{4(n-1)}\operatorname{Scal}_{g_{\mathrm{rd}}} =-\Delta_{g_{\mathrm{rd}}}+\frac{n(n-2)}4. \] The supplied record does not specify the constant normalization of $Y$. The displayed formula fixes a conventional representative; multiplying $Y$ by a positive constant does not change the monotonicity question, since it only rescales time and multiplies the scaled heat trace by a positive constant.

For every $W\in C^\infty(S^n;{\mathbb{R}})$ satisfying \[ W(x)>0\quad(x\in S^n),\qquad \int_{S^n}W^{n/2}\,{\,\mathrm{d}} v_{\mathrm{rd}} =\operatorname{vol}_{\mathrm{rd}}(S^n), \] define \[ Y_W=W^{-1/2}YW^{-1/2}, \] where the factors $W^{-1/2}$ act by multiplication. Regard $Y_W$ as the positive self-adjoint realization on $L^2(S^n,{\,\mathrm{d}} v_{\mathrm{rd}})$ of this elliptic operator initially defined on smooth functions. No boundary conditions are needed. If $\lambda_j(W)$, $j\geq1$, are its eigenvalues, repeated according to multiplicity, set \[ f_W(t)=t^{n/2}\operatorname{Tr}(e^{-tY_W}) =t^{n/2}\sum_{j=1}^{\infty}e^{-t\lambda_j(W)}, \qquad t>0. \] Question. For every integer $n\geq4$ and every weight $W$ satisfying the conditions above, is \[ f_W(t_2)\leq f_W(t_1) \qquad\text{for all }0<t_1<t_2<\infty? \] This formulates "decreasing" as nonincreasing. The supplied record does not explicitly distinguish this from strict decrease; if strict decrease is intended, the displayed inequality must instead be strict.

The supplied dated curation labels the problem open and reports that the conjecture is proved for $W=1$ and for large $t$, whereas it fails in dimension $3$ at small $t$. These are supplied source assessments, not independently verified assertions here. The record does not specify the dependence or uniformity of the large-time threshold. It also mentions a comparison question for arbitrary positive weights without supplying its mathematical formulation; no such comparison is added to the question above.

Short English statement

For each integer $n\geq4$, consider every smooth positive weight on the round sphere whose $n/2$-power has the same integral as the constant weight $1$. Must the heat trace of the corresponding weighted conformal Laplacian, multiplied by $t^{n/2}$, be nonincreasing for every positive time? The supplied word "decreasing" leaves strictness unspecified. The open label and reported special cases are supplied dated curation and require independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30001169$, title "Monotonicity of Weighted Yamabe Heat Traces," source problem code catalogue entry; supplied collection: Oberwolfach Reports Open Problems. The record lists proposed year 2009 but supplies no individual proposer attribution.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. Supplied source and literature link: https://doi.org/10.4171/owr/2009/06. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports proofs for $W=1$ and for large $t$, and reports failure in dimension $3$ at small $t$. It describes general monotonicity and comparison for arbitrary positive weights as remaining open. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied registry marks the statement corrected_verified and says its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This records the supplied verification claim; the report has not been independently consulted for this draft. No specific problem page or named contributor is supplied.