Optimality of Theorem 3 regularity at elliptic complex points

Establish whether the regularity asserted in Theorem 3, namely C^{(k-n+2)/2,\alpha/2} regularity of the Levi-flat hypersurface near a complex point whose Taylor expansion has the form given in equation (25), is optimal at every elliptic point, including umbilic elliptic points.

Background

The paper proves regularity for Bishop families of holomorphic discs near non-umbilic elliptic complex points and extends the method to certain higher-index complex points satisfying the Taylor-expansion condition in equation (25). Theorem 3 gives C^{(k-n+2)/2,\alpha/2} regularity for the resulting Levi-flat hypersurface when the boundary surface is C^{k,\alpha}.

The authors note that their proof does not apply to umbilic elliptic complex points, while prior work provides examples with lower regularity. They explicitly conjecture that the regularity in Theorem 3 is nevertheless optimal at every elliptic point, leaving the general optimality question unresolved.