Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }\ge (n-2)\frac{\pi }{2}$, 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 $\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }\ge (n-2)\frac{\pi }{2}$ and $n>2$.