Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity

TL;DR

This paper derives detailed asymptotic expansions at infinity for 2D gradient graphs with vanishing mean curvature, extending higher-dimensional results and employing refined iterative methods with spherical harmonics.

Contribution

It introduces a novel approach using iteration methods and spherical harmonic expansions to explicitly characterize asymptotic behavior in two dimensions.

Findings

Established asymptotic expansion at infinity for 2D gradient graphs

Generalized higher-dimensional results to 2D case

Provided more explicit asymptotic behavior through refined methods

Abstract

In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for minimal gradient graph on exterior domain in dimension 2. Different from the strategies for higher dimensions, instead of the equivalence of Green's function on unbounded domains, we apply a version of iteration methods from Bao--Li--Zhang [Calc.Var PDE, 52(2015), pp. 39-63] that is refined by spherical harmonic expansions to provide a more explicit asymptotic behavior than known results.