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September 23, 2025Open Access

Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow equation in R^n+1_-

Key Points

Classical solutions converge exponentially to a quadratic polynomial, with a specified convergence rate, addressing key aspects of the Lagrangian mean curvature flow.
Liouville type rigidity theorems are central to understanding the asymptotic behavior of solutions in various dimensions, particularly for n ≥ 2.
Dirichlet type problems are explored with prescribed asymptotic behavior, highlighting innovations in the applicability of the methods used.
The findings connect Liouville rigidity, asymptotic analysis, and boundary value problems, thereby enhancing the understanding of their interplay.

Abstract

This paper investigates the asymptotic behavior at infinity of ancient solutions to the Lagrangian mean curvature flow equation. Under conditions that admit Liouville type rigidity theorems, we prove that every classical solution converges exponentially to a quadratic polynomial of form τt+12x'Ax+bx+c at infinity, with an explicitly derived convergence rate. Furthermore, we investigate Dirichlet type problems with prescribed asymptotic behavior, featuring two key innovations: applicability to all dimensions n 2, and no requirement for the matrix A to be positive definite or close to a scalar multiple of the identity matrix. These results establish the relationship among Liouville type rigidity, asymptotic analysis at infinity, and boundary value problems (including Dirichlet type and initial value type).

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