Longtime existence of the Lagrangian mean curvature flow
Let \((M,J,g)\) be a Kähler-Einstein manifold, and \(\omega\) be the corresponding symplectic form. Let us denote by \(\Sigma(M)\) the space of symmetric tensors \(S(X,Y)\) on \(M\) such that \(S(JX,Y)=S(X,JY)\), \(\nabla S=0\), where \(\nabla\) is the Levi-Civita connection of \(g\). The author considers compact Lagrangian submanifolds in flat \((M,g)\) evolving by its mean curvature, and proves uniform \(C^{2,\alpha}\)-bounds in space and \(C^2\)-estimates in time for the underlying Monge-Ampere equation. Let \(F_t\) be a solution of the mean curvature flow equation \(dF/dt= \vec H\). The main result is: If \(F_0 : L \to (M,g,J)\) is a compact Lagrangian immersion into a flat manifold such that there exists a tensor \(S \in \Sigma(M)\) with \(F_0^*S(X,X)>0\) for all \(X \neq 0\), then there exist constants \(c_1,c_2>0\) such that \(\lim \sup_{t \to T} \max_{F_t(L)} | \vec H| \leq c_1\), \(\lim \sup_{t \to T} \max_{F_t(L)} | d^+ \vec H| \leq c_2\), on the maximal interval \([0,T)\), where the smooth solution \(F_t\) exists (\(d^+H = \nabla_i H^i\)). All \(F_t^*g\) are uniformly equivalent to \(F_0^*g\). If \(n=2\), or \(F_0^*\bar S(X,X)>0\) for all \(X \neq 0\), where \(\bar S(X,Y) = -S(JX,Y)\), then \(T=\infty\) and the Lagrangian submanifolds converge smoothly to a flat Lagrangian submanifold. This result implies, in part, the following corollary. Under the given assumptions, the Lagrangian submanifold can be written as a graph \((x, Du(x))\) over a flat Lagrangian subspace. If \(u: \mathbb R^n \to \mathbb R\) is a smooth strictly convex function such that all eigenvalues of Hess\((u)\) are bounded by 1 and such that \(F(x)=(x, Du(x))\) is the universal cover of a compact Lagrangian submanifold in \(\mathbb C^n = \mathbb R^n \oplus i \mathbb R^n\), then the Lagrangian mean curvature flow deforms \(L\) into a flat plane.
scientific article; zbMATH DE number 2123241