Regularity from the Guan–Sawyer argument for C^{4,\alpha} metrics

Determine whether the Guan–Sawyer argument yields a global C^{2,1}-isometric embedding into \mathbb{R}^3 for a C^{4,\alpha} metric with nonnegative Gaussian curvature when the mean curvature of the existing C^{1,1}-isometric embedding is bounded below by a positive constant.

Background

The paper proves a dichotomy for a locally convex C{1,1}-isometric immersion near an isolated zero of the Gaussian curvature whose Hessian is positive definite. In one branch, the mean curvature is bounded below by a positive constant; in the other, the immersion is shown directly to have C{2,1} regularity.

For the positive-mean-curvature branch, the authors invoke a theorem of Guan and Sawyer, which establishes smoothness under a finite-type curvature condition. They explicitly note that it is unresolved whether the same argument provides the weaker C{2,1} regularity for metrics of class C{4,\alpha}, because the Guan–Sawyer proof uses Calabi-type calculations involving third-order derivatives of the Gaussian curvature, whereas a C{4,\alpha} metric supplies only the regularity stated in the remark.

References

However, in the first alternative with the lower bound $H\geq c>0$, we are currently uncertain whether the arguments in Guan--Sawyer would yield $ \in C{2,1}$ for $g \in C{4,\alpha}$. This is because crucially relies on Calabi-type calculations as in Rios--Sawyer--Wheeden, which involves taking third-order derivatives of $K_g$.

— On the degenerate Weyl problem on isometric immersions  (2609.28905 - Li et al., 24 Sep 2026) in Remark following Theorem (Dichotomy), Section (Intrinsic sufficient condition for C^{2,1}-isometric immersions)