Geometric Harnack inequalities for level lines

Prove geometric Harnack inequalities for level lines arising from solutions of the elliptic problems considered on domains in O_C, quantifying how geometric features propagate across level sets.

Background

Beyond classical analytic Harnack inequalities, the paper’s geometric viewpoint suggests potential geometric analogues tailored to level sets. Such inequalities would control geometric variation or comparability across different parts of a level set.

The authors explicitly raise the possibility of establishing geometric Harnack-type inequalities in this setting.

References

Open questions remain:

  • Can geometric Harnack inequalities be obtained for level lines?
— Geometric Properties of Level Sets for Domains under Geometric Normal Property  (2603.30026 - Barkatou, 31 Mar 2026) in Conclusion

Although it is rather unclear what $f$ satisfy these assumptions, the simple example of $\det D2u = 1$ satisfies them.

— Geometric Viewpoint of $L^\infty$ Estimates for Linearized Monge-Ampère Equations  (2610.23848 - Rakshit et al., 16 Oct 2026) in Section 3, $L^\infty$ Estimates and Sobolev Inequality