Higher-dimensional small-height Lipschitz estimates and optimal thresholds

Establish uniform interior Lipschitz estimates for continuous viscosity solutions of the n-dimensional special Lagrangian equation in the subcritical phase range for n≥4 under the proposed small-height conditions, and determine the optimal scale-invariant critical height threshold M_crit(n,Θ) for each subcritical phase.

Background

The paper proves the sharp height threshold M=sec Θ for the three-dimensional special Lagrangian equation with subcritical phase |Θ|<π/2. It then discusses the unresolved extension to dimensions n≥4, where a one-point maximum-principle argument provides sufficient small-height bounds in part of the subcritical range, but the proposed bounds are not known to be optimal.

For example, at phase Θ=0, the discussion suggests the sufficient condition M<tan(π/[2(n−1)]), yielding M<1/√3 in dimension four and M<√2−1 in dimension five. The authors state that the sharpness of these bounds is unknown and formulate the broader problem of proving the higher-dimensional extension and determining the optimal critical height.

References

We expect these bounds to yield uniform interior Lipschitz estimates for continuous viscosity solutions, with the constants and the interior radius depending only on n, |Θ|, and M. Their sharpness is unknown. A natural problem is to establish this higher-dimensional extension and to determine the optimal height threshold M_{\mathrm{crit}(n,\Theta)} for each subcritical phase.

— Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three  (2610.01986 - Li et al., 1 Oct 2026) in Remark following the proof of Theorem 1.1 (higher-dimensional case)