Three meanings of a global ambient metric

Definitions and mathematical statement

Michael Eastwood's Problem 3 asks, without further qualification, whether there is a global ambient metric construction. The paired aid distinguishes: construction formal along the entire conformal base, an exact smooth Ricci-flat ambient metric on a domain, and a geodesically complete global extension. These are different assertions.

In the usual conformal dimension $n\geq3$, an ambient normal form for a representative $g$ is \[ \widetilde g=2\rho\,dt^2+2t\,dt\,d\rho+t^2g_\rho, \qquad t>0,\quad g_0=g. \] It is homogeneous of degree two under scaling of $t$ and restricts to the tautological metric on the cone at $\rho=0$. The Fefferman--Graham formal construction is global along the base in its appropriate formal sense. In even dimensions its obstruction and allowed order/ambiguity must be retained; a nonzero ambient obstruction prevents an exact smooth Ricci-flat construction of the desired type. Formal base-globality is therefore not unconditional existence of an exact smooth all-order metric or a complete manifold.

For an Einstein representative with \[ \operatorname{Ric}(g)=2\lambda(n-1)g, \] the known exact family reported and verified in the aid is \[ g_\rho=(1+\lambda\rho)^2g,\qquad \widetilde g=2\rho\,dt^2+2t\,dt\,d\rho +t^2(1+\lambda\rho)^2g. \] It is Ricci flat on $(0,\infty)\times M\times I_\lambda$, where the maximal connected nondegenerate normal-form interval containing $0$ is \[ I_\lambda=\begin{cases} \mathbb R,&\lambda=0,\\ (-1/\lambda,\infty),&\lambda>0,\\ (-\infty,-1/\lambda),&\lambda<0. \end{cases} \] This construction is exact over the entire base for this special class. Maximality here is only within the displayed connected nondegenerate normal-form domain, not maximality among all coordinate-independent ambient extensions.

The aid supplies an incompleteness diagnostic on that domain. With $x,\rho$ fixed, coordinate $t$-lines are affine geodesics and reach the missing endpoint $t=0$ in finite affine parameter. For $\lambda\ne0$, coordinate $\rho$-lines with $t,x$ fixed are also affine geodesics and reach $\rho=-1/\lambda$ in finite parameter. Thus this exact base-global normal-form domain is geodesically incomplete, including the $\lambda=0$ case via $t$-lines. The conclusion does not prohibit a further extension in other coordinates, nor establish a general no-complete-ambient theorem. The original word global must be specified before these special-case results can be regarded as an answer.

Short English statement

The supplied aid separates formal base-global construction, exact smooth construction and completeness. An explicit Einstein ambient metric is Ricci flat on a maximal nondegenerate normal-form interval, yet that domain is incomplete. Even-dimensional obstruction limits general exact smooth constructions; a complete extension is a separate question.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[358], Problem 3, under Michael Eastwood's contribution. Title: "Three meanings of a global ambient metric."
• AIM workshop Conformal structure in geometry, analysis, and physics, supplied reference https://aimath.org/WWN/confstruct/confstruct.pdf. The original question is exactly "Is there a global ambient metric construction?" Original remarks and literature fields are empty; this draft does not access a PDF.
• Supplied research summary invokes the Fefferman--Graham formal construction and ambient obstruction, and a known Einstein normal-form family. No full bibliographic entries or theorem numbers are supplied. Attempt 1 supplies the interval and affine-coordinate-geodesic diagnostic.
• Registry dated 14 August 2026: "partially_solved," "PARTIAL-PROGRESS," statement status "exact"; attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.