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

Jiguang Bao, Zixiao Liu

TL;DR

This work analyzes ancient solutions to the Lagrangian mean curvature flow in $\mathbb{R}^{n+1}_-$ with compactly supported forcing $f$, establishing Liouville-type rigidity that forces asymptotic behavior to a quadratic form in space plus a linear term in time, with an explicit exponential convergence rate. It also constructs a global viscosity solution with prescribed infinity behavior for all dimensions $n\ge 2$ without imposing positivity constraints on the Hessian of the quadratic term, using translated Dirichlet problems, barrier constructions, and Perron’s method. The paper then translates the asymptotics into an initial-value problem to derive a polynomial convergence rate for viscosity solutions, and, under interior regularity (bounded Hessian), upgrades to exponential convergence via linearization and fundamental solutions. These results connect rigidity phenomena, asymptotic analysis, and the existence of viscosity solutions, offering a framework applicable to a broad range of dimensions and forcing terms.

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.

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

TL;DR

This work analyzes ancient solutions to the Lagrangian mean curvature flow in with compactly supported forcing , establishing Liouville-type rigidity that forces asymptotic behavior to a quadratic form in space plus a linear term in time, with an explicit exponential convergence rate. It also constructs a global viscosity solution with prescribed infinity behavior for all dimensions without imposing positivity constraints on the Hessian of the quadratic term, using translated Dirichlet problems, barrier constructions, and Perron’s method. The paper then translates the asymptotics into an initial-value problem to derive a polynomial convergence rate for viscosity solutions, and, under interior regularity (bounded Hessian), upgrades to exponential convergence via linearization and fundamental solutions. These results connect rigidity phenomena, asymptotic analysis, and the existence of viscosity solutions, offering a framework applicable to a broad range of dimensions and forcing terms.

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 and a linear function in , 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 , 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.

Paper Structure

This paper contains 10 sections, 28 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Initial value problem without growth condition at infinity
  3. Existence of viscosity solution to Lagrangian mean curvature flow
  4. Translated Dirichlet problems
  5. Construction of subsolutions
  6. Construction of supersolutions
  7. Proof of Theorem \ref{['thm-Main3-EntireDiri']}
  8. Proof of Theorems \ref{['thm-Main1-AsymBehav']} and \ref{['thm-main5-GeneralThm']}
  9. The convergence rate for viscosity solutions
  10. Exponential convergence rate for solutions with interior estimates

Key Result

Theorem 1.1

Let $n\geq 2$, $f$ satisfy the condition $(F)$, and $u\in C^{2,1}(\mathbb R^{n+1}_-)$ be a classical solution to equ-LagFlow-Extended. Assume further that either of the following conditions holds, Then there exist $A\in\mathrm{Sym}(n)$, $b\in\mathbb R^n, c\in\mathbb R$, and $\tau:=\sum_{i=1}^n\arctan\lambda_i(A)$ such that satisfies $E(x,t)=0$ in $\mathbb R^n\times(-\infty,-T]$, and for some pos

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Theorem 2.2: Theorem 2.1 in Barles-Biton-Ley-UniqueWithoutGrowth and Theorem 1.1 in Chen-Pang-UniqueLagranMeanFlow
  • Lemma 2.3
  • proof
  • ...and 44 more