5.7 (Agol) — Renormalized volume as a metric

Definitions and mathematical statement

Let $S$ be a surface in the source's quasi-Fuchsian setting, and let $\mathcal T(S)$ be its Teichmüller space of marked conformal structures. The renormalized volume of the quasi-Fuchsian manifold determined by the pair of conformal boundary structures defines the source's function \[\rho:\mathcal T(S)\times\mathcal T(S)\to{\mathbb{R}}.\] The surface type, opposite-boundary orientation identification and renormalized-volume normalization must be fixed by the surrounding construction; they are not reproduced in the excerpt.

Agol asks whether $\rho$ itself is a metric, meaning that for all $X,Y,Z\in\mathcal T(S)$, \[\rho(X,Y)\geq0,\qquad \rho(X,Y)=0\Longleftrightarrow X=Y,\] \[\rho(X,Y)=\rho(Y,X),\qquad \rho(X,Z)\leq\rho(X,Y)+\rho(Y,Z).\] The equality criterion concerns points of Teichmüller space, including markings, rather than merely the underlying unmarked Riemann surfaces. The question is about this two-variable function as a distance; no square root, path-length construction or mixed derivative is substituted. Additive normalizations in a volume matter to these axioms, so even the diagonal-value convention must be identified.

The supplied summary reports a relation between renormalized volume and a Kähler potential for the Weil--Petersson geometry, citing Krasnov--Schlenker. A differential tensor or symplectic form recovered from a potential is a different object from the stated two-point distance. The summary's phrase "as a candidate distance via its second mixed derivative" is therefore not used to redefine $\rho$. This draft records the exact metric-axiom question without inferring its answer from the adjacent potential statement.

Short English statement

Does the given renormalized-volume function satisfy all distance axioms on marked Teichmüller space? The question concerns its values directly, with its normalization fixed, rather than the Weil--Petersson tensor obtained by differentiation.

Sources

• Local UnsolvedMath registry, permanent ID $10900032$, exact source code catalogue entry.
• Agol, attribution in Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 5.7, PDF page 4: https://arxiv.org/pdf/1512.04620#page=4; abstract https://arxiv.org/abs/1512.04620. Registry access 2026-07-29; source-tex extraction recorded as checked against rendered PDF text.
• Krasnov--Schlenker and the survey The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds, https://arxiv.org/abs/0907.2590, as cited in the supplied summary for the Kähler-potential context.