Optimal stability exponent in dimension two

Determine the optimal exponent governing the quantitative rigidity estimate dist(M,H_c) lesssim delta(M)^alpha for two-dimensional pinched CartanHadamard manifolds, where the paper currently establishes only the exponent alpha=1/2.

Background

For a pinched CartanHadamard manifold with sectional curvature bounded above by a negative constant c and below by c_0, the paper defines a HeisenbergPauliWeyl deficit delta(M) using a Gaussian profile and a curvatureweighted distance dist(M,H_c) from the hyperbolic model space of curvature c. The quantitative rigidity theorem proves that dist(M,H_c) is bounded linearly by delta(M) when ngeq3, while in dimension two it obtains the weaker bound dist(M,H_c) lesssim delta(M){1/2}.

The authors explicitly state that it is unknown whether the exponent 1/2 in dimension two is optimal. Thus, the unresolved problem is to determine the largest, or otherwise sharp, exponent that can replace 1/2 in the two-dimensional stability estimate.

References

At present, it is not known whether this exponent is optimal. The appearance of the exponent 1/2 stems from the underlying proof, in particular technical in nature, and it remains an interesting open problem to determine the optimal exponent in dimension two.

— Gromov-Hausdorff Stability and Rigidity of manifolds via Heisenberg-Pauli-Weyl Uncertainty Principle  (2609.01463 - Bhakta et al., 1 Sep 2026) in Remark immediately following Theorem (quantitative rigidity result), Section 1, subsection on CartanHadamard manifolds