Abstract
Let
F_{n}
: (Σ,
h_{n}
)
\to
\mathbb{C}^{2}
be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics
\{
h_{n}
\}
converges smoothly to a Riemannian metric h. We show that a subsequence of
\{
F_{n}
\}
converges smoothly to a branched conformally immersed Lagrangian self-shrinker
F_{\infty}
: (Σ, h)
\to
\mathbb{C}^{2}
. When the area bound is less than 16π, the limit
{F_{\infty}}
is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence
h_{n}
\to
h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.
Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow