Some counterexamples for the special lagrangian curvature equation

We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant positive Gauss curvature to construct an explicit Lipschitz viscosity solution which is not $C^1$. Second, still in dimension two, we construct a sequence of smooth admissible solutions on a fixed rectangle with uniformly bounded $C^1$-norm but unbounded curvature at one point; furthermore, we show that any uniform $C^{1,\beta}$ estimate fails for $\beta > 1/3$. Third, in dimension three and in the subcritical phase, we construct a Mooney-Savin type Lipschitz viscosity solution whose gradient has a jump discontinuity across an analytic surface. These examples demonstrate the sharpness of the recent a priori estimates by Qiu and Zhou, revealing that their structural assumptions-convexity and the critical phase-are strictly necessary.