Prescribing the Maslov form of Lagrangian immersions
Let \(({\mathbb R}^{2n},J,\omega)\) be the Euclidean space equipped with the standard symplectic structure \(\omega\) and complex structure \(J\). An immersion \(F:L\to{\mathbb R}^{2n}\), where \(L\) is an \(n\)-dimensional manifold, is called Lagrangian, if \(F^*\omega=0\). Equivalently \(L\) is Lagrangian, if \(J\) gives an isomorphism between the tangent and the normal bundle of \(L\). It follows that normal variations of a Lagrangian immersion can be described in terms of \(1\)-forms \(\theta\in\Omega^1(L,{\mathbb R})\). If \(\theta\) is closed, then the variation preserves the Lagrangian condition. In case of an exact \(1\)-form \(\theta=d\alpha\), the corresponding flow is called Hamiltonian. \smallskip The author studies the quasilinear parabolic system \[ (*)\quad d F/dt=-g^{ij}(H_ i-m_ i)\nu_ j, \] where \(H=H_ idx^ i\) denotes the mean curvature form of the evolving immersions and \(m=m_ idx^ i\in\Omega^1(L,{\mathbb R})\) is a fixed, time independent \(1\)-form on \(L\) such that for the initial immersion \([H]=[m]\). \(g^{ij}\) denotes the inverse of the induced metric on \(L\) and \(\nu_j\) is a local frame in the normal bundle. It then follows that \((*)\) is a Hamiltonian flow for all \(t\) for which a smooth solution of \((*)\) exists and that the evolving submanifolds stay Lagrangian.\smallskip The author shows that the flow exists for all \(t\in[0,\infty)\) and that the Lagrangian immersions converge to a Lagrangian submanifold with prescribed Maslov form \(M=m/\pi\) provided the second fundamental form of the evolving Lagrangian hypersurfaces is uniformly bounded and the induced metrics are uniformly equivalent for all \(t\). In the second part of the paper, the author derives sharp results for the \(1\)-dimensional situation of evolving starshaped curves. In particular, he obtains the following results: Theorem 1.3: Assume that \(R : S^ 1\to{\mathbb R}\) is a smooth positive function and \(\gamma_ 0\) the initial curve given by the immersion \(F:S^ 1\to{\mathbb R}^2\) \[ F(\phi):=(r(\phi)\cos{(k\phi)},r(\phi)\sin{(k\phi)}), \] where \(k\) is a positive integer. Further assume that \(f:S^ 1\to{\mathbb R}\) is a smooth function such that \[ | \arctan{(r'/(kr))}| <\pi/2-\text{osc}(f). \] If we set \(m=(f-k\phi)'d\phi\), then \((*)\) admits a smooth solution for \(t\in[0,\infty)\) and the curves smoothly converge to a limit curve with curvature form \(H=m\). Theorem 1.4: Assume \(u,f:{\mathbb R}\to{\mathbb R}\) are smooth periodic functions with the same period such that the initial curve \(F:{\mathbb R}\to{\mathbb R}^2\) given by the graph of \(u\), i.e. \[ F(x):=(x,u(x)) \] satisfies \[ | \arctan{(u')}| <\pi/2-\text{osc}(f). \] Then, if we set \(m=df\), the flow \((*)\) admits a smooth solution for \(t\in[0,\infty)\) and the curves smoothly converge to a limit curve with curvature form \(H=df\). We remark that the quantities \(\arctan{(r'/(kr))}\) and \(\arctan{(u')}\) are related to the normal angle (or Lagrangian angle) \(\alpha\) by \(\alpha=\arctan{(r'/(kr))}-k\phi\) resp. by \(\alpha=\arctan{(u')}\). The results are illustrated by a number of figures. In case \(m=0\), \((*)\) reduces to the Lagrangian mean curvature flow which has been studied in [\textit{M. T. Wang}, Math. Res. Lett. 8, 651--661 (2001; Zbl 1081.53056)] and in details by the author [\textit{K. Smoczyk}, The Lagrangian mean curvature flow (German) Leipzig: Univ. Leipzig (Habil.), 102 S. (2000; Zbl 0978.53124 ) and Math. Z. 240, 849--883 (2002; Zbl 1020.53045)].
scientific article; zbMATH DE number 1786838