Lagrangian $L$-stability of Lagrangian Translating Solitons

Abstract

In this paper, we prove that any Lagrangian translating soliton is Lagrangian L-stable.

$f$-minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

“…The stability of soliton solutions to mean curvature flow under certain weighted volume functional was first studied by Colding-Minicozzi [7] for shrinking solitons in hypersurface case, and generalized to higher codimensional case by Andrews-Li-Wei [1], Arezzo-Sun [2], and Lee-Lue [16], but note that their functional ("entropy") is different from the f -volume functional. For stability of translating solitons, the second variation formula for translating hypersurfaces under f -volume was obtained by Xin in [35], Shahriyari [28] studied the stability of graphical translating surfaces in R 3 , and Yang [38] and Sun [30] studied the Lagrangian translating solitons. In particular, Yang [38] proved that every Lagrangian translating soliton is Hamiltonian f -stable, and Sun [30] showed that they are actually Lagrangian f -stable.…”

“…For stability of translating solitons, the second variation formula for translating hypersurfaces under f -volume was obtained by Xin in [35], Shahriyari [28] studied the stability of graphical translating surfaces in R 3 , and Yang [38] and Sun [30] studied the Lagrangian translating solitons. In particular, Yang [38] proved that every Lagrangian translating soliton is Hamiltonian f -stable, and Sun [30] showed that they are actually Lagrangian f -stable.…”

The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ

“…Translating surfaces characterize the type II finite singularity of mean curvature flow in Euclidean space (see [2], [3] and [13]). Some geometric properties were investigated in [1,20,10,4,19] etc.…”