Hamiltonian L-stability of Lagrangian translating solitons

Abstract

Abstract. In this paper, we compute the first and second variation formulas for the F-functional of translating solitons and study the Hamiltonian L-stability of Lagrangian translating solitons. We prove that any Lagrangian translating soliton is Hamiltonian 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.…”

Lagrangian $L$-stability of Lagrangian Translating Solitons

“…In this section, we will recall some results for the first variation and second variation formulas. Since the proofs can be found in Section 4 of [1] with f = T, x , where we dealt with more general cases (see also [14]), we omit the details here.…”

“…A translating soliton Σ n in C n is called a Lagrangian translating soliton if it is also a Lagrangian submanifold of C n . In [14], L. Yang proved that any Lagrangian translating soliton is Hamiltonian L-stable. In this paper, we prove that it is in fact Lagrangian L-stable: Theorem 1.1.…”