Two-channel metric variation and a transverse Einstein certificate

Definitions and mathematical statement

Thomas Branson's Problem 1a asks what information about a metric functional, already understood along conformal changes, can be obtained by moving across conformal classes. For an integral of a local invariant it suggests computing an anti-conformal variation; for a nonlocal spectral invariant such as a functional determinant, computing that variation is itself part of the challenge. The source does not fix a functional, operator, dimension, or precise meaning of anti-conformal.

The supplied aid adopts the following infinitesimal interpretation. Let $M$ be a closed smooth manifold of dimension $n$, $g$ a Riemannian metric, and $F$ a differentiable functional having a smooth $L^2$ gradient $T_g$, so that for every smooth symmetric covariant two-tensor $h$, \[ D F_g(h)=\int_M\langle T_g,h\rangle_g{\,\mathrm{d}} V_g. \] Infinitesimal conformal changes are $h=2ug$, $u\in C^\infty(M)$, and their $L^2$-orthogonal complement consists of trace-free tensors. Write \[ T_g^0=T_g-\tfrac1n(\operatorname{tr}_gT_g)g. \] Then the reported two responses are \[ D F_g(2ug)=2\int_Mu\operatorname{tr}_gT_g{\,\mathrm{d}} V_g, \qquad D F_g(h)=\int_M\langle T_g^0,h\rangle{\,\mathrm{d}} V_g \quad(\operatorname{tr}_gh=0). \] Thus conformal information records the trace of the gradient, and the chosen anti-conformal information records its trace-free part. This is one working interpretation of the workshop wording.

If $F$ is invariant under diffeomorphisms, the aid reports $\operatorname{div}_gT_g=0$. A transverse-traceless tensor is a smooth $h$ with both $\operatorname{tr}_gh=0$ and $\operatorname{div}_gh=0$. Its no-loss test states \[ \left[D F_g(2ug)=0\ \forall u, \quad D F_g(h)=0\ \forall h\text{ transverse-traceless}\right] \quad\Longleftrightarrow\quad D F_g(h)=0\ \forall h. \] The hypotheses of diffeomorphism invariance and a smooth $L^2$ gradient remain essential. The assertion does not identify all trace-free tensors with transverse-traceless tensors.

For the Einstein--Hilbert functional $F(g)=\int_M R_g{\,\mathrm{d}} V_g$ in dimension $n\ge3$, using covariant metric variation, the aid specializes to \[ T_g=-\operatorname{Ric}_g+\tfrac12R_gg,\qquad T_g^0=-\operatorname{Ric}_g^0. \] Consequently vanishing on every trace-free variation is equivalent to $\operatorname{Ric}_g^0=0$, and the trace-free direction $h=\operatorname{Ric}_g^0$ gives \[ D F_g(\operatorname{Ric}_g^0) =-\|\operatorname{Ric}_g^0\|_{L^2(g)}^2<0 \quad\text{if }\operatorname{Ric}_g^0\not\equiv0. \] This is a trace-free descent certificate. It is not automatically a transverse-traceless direction: in these conventions $\operatorname{div}_g\operatorname{Ric}_g^0=(\tfrac12-\tfrac1n){\,\mathrm{d}} R_g$. The aid supplies no anti-conformal determinant formula; nonlocal spectral regularity, operator choice, zero modes, and determinant conventions remain to be specified before that part of the research request can be made precise.

Short English statement

The source asks how metric functionals change across conformal classes. The aid interprets these directions as trace-free variations and separates the gradient's trace and trace-free responses. For diffeomorphism-invariant functionals it gives a full-stationarity test using conformal and transverse-traceless directions, while its Einstein--Hilbert descent direction is trace-free and need not itself be transverse-traceless.

Sources

• Local UnsolvedMath registry, permanent ID 20002017, exact source code catalogue entry; canonical aim-geometry-notes.json, notes[354], tag problem; AIM Workshop Problem Lists, Geometry. Created and updated 14 August 2026.
• AIM workshop Conformal structure in geometry, analysis, and physics, 12--16 August 2003, subsection "Thomas Branson. Anti-conformal perturbations", Problem 1a; https://aimath.org/WWN/confstruct/confstruct.pdf. This attribution and location are supplied by the verification; the PDF was not accessed.
• Attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Candidate: conformal/transverse-traceless stationarity lemma and Einstein--Hilbert trace-free descent certificate. No separate authoritative theorem citation is supplied.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.