[Paper Review] Collapsing of Calabi-Yau manifolds and special lagrangian submanifolds

Yuguang Zhang|ArXiv.org|Nov 5, 2009

Geometric Analysis and Curvature Flows18 references5 citations

TL;DR

This paper establishes a bidirectional relationship between the existence of special Lagrangian submanifolds and the collapsing of Ricci-flat Calabi-Yau manifolds in the Gromov-Hausdorff sense. It proves that regions with uniformly small special Lagrangian volumes must collapse, and conversely, that sufficiently collapsed regions in Calabi-Yau manifolds admit special Lagrangian fibrations, providing geometric evidence for the refined SYZ conjecture.

ABSTRACT

In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite direction,it is shown that the existence of special lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi-Yau manifolds.

Motivation & Objective

• To investigate the geometric relationship between special Lagrangian submanifolds and the collapsing of Ricci-flat Calabi-Yau manifolds.
• To provide evidence for the refined SYZ conjecture by linking the existence of special Lagrangian fibrations to metric collapse.
• To establish quantitative volume estimates that imply collapsing when special Lagrangian volumes shrink.
• To extend results from K3 surfaces to general Calabi-Yau manifolds using Gromov-Hausdorff convergence and Ricci-flat geometry.

Proposed method

• Utilizes Gromov-Hausdorff convergence of sequences of Ricci-flat Calabi-Yau manifolds to analyze limit spaces under collapsing.
• Applies the pointed Gromov compactness theorem to extract convergent subsequences of pointed manifolds.
• Employs the Bishop-Gromov volume comparison theorem to derive uniform lower bounds on injectivity radius and volume growth.
• Uses calibrated geometry: applies the calibration form Θ to bound the volume of special Lagrangian submanifolds via ∫_A Θ.
• Applies Theorem 2.6 on volume comparison in manifolds with bounded curvature to derive contradiction when volumes are too small.
• Uses smooth convergence of pullback metrics F_k,D^*g_k → g_∞ on compact subsets to analyze local geometry in the limit.

Experimental results

Research questions

• RQ1 Does the existence of special Lagrangian submanifolds with arbitrarily small volume imply metric collapse in Calabi-Yau manifolds?
• RQ2 Can special Lagrangian fibrations be constructed in regions of Ricci-flat Calabi-Yau manifolds that are sufficiently collapsed?
• RQ3 To what extent does the volume of calibrated submanifolds control the geometry of the ambient space?
• RQ4 How does the refined SYZ conjecture manifest geometrically through volume and curvature constraints?

Key findings

• If the volume of a special Lagrangian submanifold representing a fixed homology class A satisfies ∫_A Θ < δ for small δ, then the volume of the unit ball in the ambient manifold is bounded by ε, implying collapse.
• When ∫_A Θ_k → 0 along a sequence of Calabi-Yau manifolds, the volume of the unit ball B_g_k(p_k,1) tends to zero, implying Gromov-Hausdorff collapse to a lower-dimensional space.
• The limit space (Y,d_Y) has Hausdorff dimension m, but the singular set S_Y has dimension < m−1, and the regular part is Ricci-flat and smooth.
• The existence of a calibrated fibration with shrinking fiber volumes forces the ambient manifold to collapse, as shown by contradiction via volume comparison.
• The result extends to G_2 and Spin(7) manifolds due to their Ricci-flat Einstein structure.
• The proof establishes a uniform lower bound on injectivity radius in the limit, which contradicts the shrinking volume of calibrated submanifolds when k is large.