Calabi-Yau流形中的Maslov类与极小性

On log minimality of weak K-moduli compactifications of Calabi-Yau varieties

For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical K-trivial) degenerations. Although weak K-moduli is not unique in general, they always contain a unique partial compactification (K-moduli). Our main theorem is the log minimality of their normalizations, under some conditions. Partially to confirm that known examples satisfy the conditions, we also include an appendix on the algebro-geometric reconstruction of Kul…

Generalized Lagrangian mean curvature flows in almost Calabi-Yau manifolds

In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian cones for generalized Lagrangian mean curvature flow with zero-Maslov class in almost Calabi-Yau manifold. As a corollary, there is no finite time Type-I singularity for such a flow.

Lagrangian Flows, Maslov Index Zero and Special Lagrangians

We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian cycles (possibly with differing phases). We use the flow of cycles to obtain the fact that special lagrangian cycles generate the part of the lagrangian homology which lies in the image of the Hurewicz homomorphism. We also establish a weak version of a conjecture of Thomas-Yau r…

Neck pinches along the Lagrangian mean curvature flow of surfaces

Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.

辛流形和Kähler流形中余迷向子流形的几何

Geometry of coisotropic submanifolds in symplectic and Kähler manifolds

本文的第一个目的是将边界位于拉格朗日子流形上的开黎曼曲面映射的著名的Maslov指标推广到边界位于辛流形中的余迷向子流形上的映射。为此，我们首先定义了上各向同性Grassmanian的{ \it Maslov环}的概念及其指标.然后引入了余迷向子流形的{ \it横Maslov丛}和{ \it可分次余迷向子流形}的概念。然后定义了{ \it分次共迷向子流形}和边界位于这类分次共迷向子流形上的映射的{ \it共迷向Maslov指数}，对于拉格朗日子流形，它退化为圆盘映射的标准Maslov指数.第二个目的是研究Kähler流形中余迷向子流形的几何。引入了余迷向子流形的叶向中曲率形式和横典范丛的概念，并研究了它们的各种几何性质。最后我们结合这些定义的概念{ \it特殊共迷向子流形}的情况下的Calabi-Yau流形，并证明各种后果，其性质的共迷向Maslov指标。

Remarks on invariants of hamiltonian loops

In this note the interrelations between several natural morphisms on the $π_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (non-linear) Maslov index of loops of quantomorphisms of prequantizations of $\C P^n$ and the Calabi-Weinstein invariant is shown, settling affirmatively a conjecture by A. Givental. We also prove the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds, as suggested by C. Woodward. Finally, a family of generalized action-Maslov inv…