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Register a new userLagrangian Flows, Maslov Index Zero and Special Lagrangians
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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 regarding lagrangian mean curvature flow.
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We review results about $G_2$-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.
We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.
We construct families of imaginary special Lagrangian cylinders near transverse Maslov index $0$ or $n$ intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold. Hence, we obtain geodesics of open positive Lagrangian submanifolds near such intersection points. Moreover, this result is a first step toward the non-perturbative construction of geodesics of closed positive Lagrangian submanifolds. Also, we introduce a method for proving $C^{1,1}$ regularity of geodesics of positive Lagrangians at the non-smooth locus. This method is used to show that $C^{1,1}$ geodesics of positive Lagrangian spheres persist under small perturbations of endpoints, improving the regularity of a previous result of the authors. In particular, we obtain the first examples of $C^{1,1}$ solutions to the positive Lagrangian geodesic equation in arbitrary dimension that are not invariant under isometries. Along the way, we study geodesics of positive Lagrangian linear subspaces in a complex vector space, and prove an a priori existence result in the case of Maslov index $0$ or $n.$ Throughout the paper, the cylindrical transform introduced in previous work of the authors plays a key role.
We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flows such as coupled mean curvature/Kaehler-Ricci flow in the sense of Smoczyk as a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm-Etnyre-Sullivan. This proves part of a conjecture of Joyce. We give a lower bound on the time for which the Floer cohomology is invariant under the (forward or backwards) flow, if it exists.
The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy-Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah-Floer conjecture.
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