Normalized least-Lipschitz geometry beyond the hyperbolic locus

Definitions and mathematical statement

D. Dumas asks how the least-Lipschitz function behaves for metrics beyond the hyperbolic locus, such as negatively curved Riemannian metrics or their closure. The displayed source omits a homotopy restriction on maps. Without one, constant maps make the infimum zero. In the marked interpretation used by the aid, for metrics on a fixed closed oriented manifold $M$ define \[ L(g,h)=\inf_{\varphi\simeq\operatorname{id}_M}\operatorname{Lip}(\varphi:(M,g)\to(M,h)). \] The original Teichmüller setting concerns surfaces; some formulas below make sense in dimension $d$ more generally. Maps need not be diffeomorphisms for the infimum as written, and the marking restriction is explicit.

For positive-definite metrics on compact $M$, the aid asserts attainment of the infimum and \[ L(g,k)\leq L(g,h)L(h,k),\qquad L(g,h)\geq\sup_\alpha\frac{\ell_h(\alpha)}{\ell_g(\alpha)},\qquad L(g,h)^d\operatorname{Vol}_g(M)\geq\operatorname{Vol}_h(M), \] where $\alpha$ ranges over the relevant nontrivial free homotopy classes. The volume inequality uses degree one. The summary reports existing work showing that equality with marked-length stretch, familiar for hyperbolic metrics, can fail in variable negative curvature; no example or full citation is supplied.

The proposed normalization is \[ \widehat L(g,h)=L(g,h)\left(\frac{\operatorname{Vol}_g(M)}{\operatorname{Vol}_h(M)}\right)^{1/d}. \] It is at least $1$, submultiplicative, and unchanged by independently scaling $g$ and $h$: $L(a^2g,b^2h)=(b/a)L(g,h)$ while the volume factor cancels $b/a$. Thus $\widehat d(g,h)=\log\widehat L(g,h)$ is a nonnegative asymmetric metric kernel. The claimed zero-distance rigidity identifies metrics by a homothety and an isometry in the prescribed identity homotopy class. For closed negatively curved surfaces, this gives the reported asymmetric metric modulo $\operatorname{Diff}_0$ and homothety. In higher dimensions, identifying identity-homotopic diffeomorphisms with $\operatorname{Diff}_0$ needs an additional isotopy hypothesis and is not silently inferred.

For a uniform tensor-comparison distance, define \[ \rho(g,g')=\inf\{r\geq0:e^{-2r}g\leq g'\leq e^{2r}g\}. \] The aid's quantitative stability statements are \[ |\log L(g',h')-\log L(g,h)|\leq\rho(g,g')+\rho(h,h'), \] \[ |\log\widehat L(g',h')-\log\widehat L(g,h)| \leq2\left(\rho(g,g')+\rho(h,h')\right). \] On a fixed-volume slice, the normalized logarithm equals the unnormalized one, so the first constant applies. These bounds provide continuous extension of the functionals to positive-definite $C^0$ metrics in the corresponding uniform topology. They do not assert an extension across degenerate or collapsed metrics, and metric separation on a nonsmooth closure is not concluded merely from continuity. The question's "closure" otherwise remains unspecified.

Short English statement

Interpret the omitted maps as marking-preserving ones; otherwise the infimum is zero. The supplied aid gives least-Lipschitz bounds, volume normalization and quantitative stability beyond hyperbolic metrics, while leaving degenerate closures and exact marked-length equality outside its scope.

Sources

• Local UnsolvedMath registry, permanent ID $20003122$, source code catalogue entry; canonical aim-topology-notes.json, notes[209].
• AIM Problems on Thurston Metric, Problem 5.4, attributed to D. Dumas, workshop Lipschitz metric on Teichmueller space: https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The supplied verification says the displayed PDF infimum contains no map restriction; the identity-homotopy condition is a conservative marked-space reconstruction. Statement status is reconstructed_unverified; original remarks and literature field are empty.
• The aid mentions existing variable-negative-curvature failures of marked-length equality without authoritative complete citations or explicit examples.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its normalization, rigidity and stability assertions are machine-generated aid, not peer reviewed, and require independent verification before citation.