A Yamabe stability model for conformal variational problems

Definitions and mathematical statement

Problem 1c asks to "Study variational problems arising from conformally invariant problems." This is a research agenda without a specified functional, domain, or conclusion. The aid places it under Thomas Branson's anti-conformal perturbations heading and identifies the following "Michael Eastwood" line as the next participant block, not attribution of this problem.

Its model is the Yamabe variational problem. Let $(M,g)$ be a closed Riemannian manifold of dimension $n\ge3$, take the nonnegative Laplacian $\Delta_g=-\operatorname{div}_g\nabla_g$, and set \[ a_n=\frac{4(n-1)}{n-2},\quad p=\frac{2n}{n-2},\quad L_g=a_n\Delta_g+R_g, \quad Y_g(u)=\frac{\int_M(a_n|\nabla u|^2+R_gu^2){\,\mathrm{d}} V_g} {(\int_M|u|^p{\,\mathrm{d}} V_g)^{2/p}},\qquad u>0. \] For $\widehat g=w^{4/(n-2)}g$, $w>0$, the aid reports \[ L_{\widehat g}\phi=w^{-(p-1)}L_g(w\phi),\qquad Y_{\widehat g}(\phi)=Y_g(w\phi). \] The positive critical-point equation is $L_gu=\kappa u^{p-1}$, with $\kappa=\int_MuL_gu{\,\mathrm{d}} V_g/\int_Mu^p{\,\mathrm{d}} V_g$.

Suppose $\operatorname{Vol}_g(M)=1$ and $R_g\equiv R$ is constant, so $u=1$ is a normalized critical point. Write \[ E_g(u)=\int_M(a_n|\nabla u|^2+Ru^2){\,\mathrm{d}} V_g, \qquad C_g(u)=\int_Mu^p{\,\mathrm{d}} V_g. \] For the constraint $C_g=1$, the multiplier in $E_g-\mu(C_g-1)$ is $\mu=2R/p$. Its constrained Hessian on $\int_Mv{\,\mathrm{d}} V_g=0$ is reported as \[ \mathcal H(v,v)=2\int_M\left(a_n|\nabla v|^2-(p-2)Rv^2\right){\,\mathrm{d}} V_g. \] Since $(p-2)/a_n=1/(n-1)$, for connected $M$ the exact infinitesimal nonnegative-stability criterion is \[ \lambda_1(\Delta_g)\ge\frac{R}{n-1}, \] where $\lambda_1$ is the first positive eigenvalue. Equality directions are the mean-zero eigenspace with eigenvalue $R/(n-1)$. If $R\le0$ and $M$ is connected, the Hessian is strictly positive on nonzero mean-zero variations. These are Hessian assertions, not a claim of strict nonlinear local minimality in every degenerate case.

If $M$ has $b_0(M)$ components but only one total-volume constraint, componentwise constants satisfying the global mean-zero condition form a $b_0(M)-1$-dimensional subspace. At $R=0$ these are exactly the additional zero modes; at $R>0$ they are negative directions; at $R<0$ they are positive. For nonconstant componentwise modes the same spectral threshold must be checked on each component. Thus connected stability cannot be transferred unchanged to a disconnected manifold.

The aid also invokes the Lichnerowicz--Obata sphere interpretation. In the explicit positive Einstein setting $\operatorname{Ric}_g=(n-1)kg$, $k>0$, one has $R=n(n-1)k$ and the relevant bound is $\lambda_1\ge nk$, with equality in the connected case characterized by the round sphere of curvature $k$. This interpretation needs its Ricci-curvature hypothesis; constant scalar curvature alone does not supply that eigenvalue bound. The model illustrates one precisely normalized conformal variational problem rather than resolving the full agenda.

Short English statement

The source gives a broad agenda rather than one theorem. The aid uses the Yamabe quotient to make a model precise, including conformal covariance, the critical equation, and a coefficient-checked Hessian threshold. It retains the extra component-transfer modes when the manifold is disconnected and avoids attributing the problem to the following participant heading.

Sources

• Local UnsolvedMath registry, permanent ID 20002019, exact source code catalogue entry; canonical aim-geometry-notes.json, notes[356], tag problem; AIM Workshop Problem Lists, Geometry, Conformal structure in geometry, analysis, and physics, Problem 1c. Created and updated 14 August 2026.
• Source URL https://aimath.org/WWN/confstruct/confstruct.pdf. The aid assigns Problem 1c to Thomas Branson's anti-conformal perturbations block and the following Michael Eastwood line to the next block. This is supplied contextual evidence; the PDF was not accessed.
• The aid invokes the Yamabe quotient and Lichnerowicz--Obata sphere interpretation, without full bibliographic entries or theorem locations.
• Attempt 1: partial_result, problem status at run open, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Candidate: normalization diagnostic, multiplier, Hessian threshold, and disconnected corrections. These are dated source labels, not certification.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.