On counterexamples to interior C² estimates for Monge-Ampère type equations

We modify Pogorelov’s classic construction to demonstrate the absence of a priori C² C 2 estimates for the equations (D² u ± Du ⊗ Du) = f(x) det ( D 2 u ± D u ⊗ D u ) = f ( x ) in dimension n ≥ 3 n ≥ 3 . We construct a sequence of solutions z_ε z ε with second derivatives blowing up at the origin as ε → 0 ε → 0 , while the corresponding right-hand sides f_ε f ε admit uniform C² C 2 estimates. Specifically, the counterexamples are given by z_ε (x₁, , xₙ) = (1+x₁²)(1+x₂²)(ε ² + η ²)α /2, z ε ( x 1 , ⋯ , x n ) = ( 1 + x 1 2 ) ( 1 + x 2 2 ) ( ε 2 + η 2 ) α / 2 , where η = √x₃² + + xₙ² η = x 3 2 + ⋯ + x n 2