Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow in $\mathbb R^{n+1}_-$

TL;DR

This paper studies the long-term behavior of ancient solutions to the Lagrangian mean curvature flow, proving convergence to a quadratic-linear form at infinity and establishing the existence of global viscosity solutions with prescribed asymptotics.

Contribution

It introduces new methods to prove convergence and existence of solutions without restrictive conditions on the Hessian, applicable in all dimensions n≥2.

Findings

Solutions converge at infinity to a quadratic polynomial plus a linear function.

Existence of global viscosity solutions with prescribed asymptotic behavior is established.

Results hold for all dimensions n≥2 without positive definiteness constraints.

Abstract

This paper investigates the asymptotic behavior at infinity of ancient solutions to the Lagrangian mean curvature flow. Under conditions that admit Liouville type rigidity theorems, we prove that every classical solution converges at infinity to the sum of a quadratic polynomial in $x$ and a linear function in $t$, with an explicitly derived exponential rate of convergence. As a critical part of the proof framework of this paper, we establish the existence of a global viscosity solution with prescribed asymptotic behavior at infinity, featuring two key innovations: (i) applicability to all dimensions $n\geq 2$, and (ii) no requirement that the Hessian matrix of the prescribed quadratic term be positive definite or close to a scalar multiple of the identity matrix. These results establish the relationship between Liouville type rigidity, asymptotic analysis at infinity, and the existence of viscosity solutions.