We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|\Theta|\geq (n-2)\tfrac{ \pi}{ 2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^{ \alpha}$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{ 2,\alpha}$ estimates by exponentiating the arctangent operator into a concave one when $|\Theta|\geq (n-2)\tfrac{ \pi}{ 2}$ and $n>2$.
Categories: math.AP math.DG