Rigidity and uniruling for Lagrangian submanifolds (0808.2440v1)

Abstract: This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

Essay on "Rigidity and Uniruling for Lagrangian Submanifolds"

The paper "Rigidity and Uniruling for Lagrangian Submanifolds" by Paul Biran and Octav Cornea provides a detailed exploration of the topology of monotone Lagrangian submanifolds within symplectic manifolds through a combination of quantum homology and various algebraic structures related to Lagrangian submanifolds. This investigation is grounded in understanding the interaction between ambient symplectic topology and the topology of the Lagrangians themselves.

Main Contributions and Results

The authors establish significant relationships between the properties of monotone Lagrangian submanifolds $L$ and ambient symplectic manifolds $M$ by utilizing the quantum homology of $M$ and the quantum structures associated with $L$. They reveal that monotone Lagrangians demonstrate structural rigidity properties, which are especially pronounced when the ambient manifold $M$ contains a substantial number of genus-zero pseudo-holomorphic curves.

Key Results:

1. Floer Homology Dichotomy:
   • The paper establishes that monotone Lagrangian submanifolds are either "narrow" or "wide." Specifically, if a Lagrangian is narrow, it means its Floer homology vanishes, and if it is wide, its Floer homology is isomorphic to the singular homology of $L$.
2. Uniruling and Width Constraints:
   • If an ambient manifold is uniruled by spheres, then Lagrangian submanifolds tend to be uniruled by disks of lower area, providing constraints on the Lagrangian width. Particularly, the width of narrow monotone Lagrangians is universally bounded.
   • The paper further establishes strong uniruling results for non-narrow Lagrangians in point invertible symplectic manifolds, with applications indicated for $\mathbb{C}P^n$.
3. Spectral Invariants:
   • The work also relates the rigidity of Lagrangians to spectral invariants. The paper generalizes known spectral invariant techniques and applies them to deduce new intersection results for Lagrangians by showing specific bounds and relations in the context of symplectic geometry.
4. Examples and Computations:
   • The authors provide explicit calculations for various examples, such as Lagrangians in $\mathbb{C}P^n$ with specific homology properties and the Clifford torus. These examples elucidate the theoretical findings and their applications to known geometrical objects.

Theoretical Implications

The theoretical implications of this work are vast. The results on narrowness and wideness, alongside the tools developed, enable a nuanced understanding of Lagrangian behavior in symplectic manifolds. The new uniruling results suggest refined classifications of symplectic manifolds based on the rigidity and flexibility of Lagrangian submanifolds they can harbor. The methodology provides a bridge between ambient symplectic topology and the internal topology of Lagrangians, showing how symplectic embedding constraints affect Lagrangian topology.

Practical Implications and Future Directions

From a practical perspective, this research advances our methods for determining the rigidity and flexibility of symplectic embeddings, crucial in areas like Hamiltonian dynamics and mirror symmetry. Potential future work could extend these methods to non-monotone settings or analyze how these structural properties influence symplectic field theory or Fukaya categories. The conjectured uniqueness, or lack thereof, of Lagrangian types in specific settings, such as $\mathbb{R}P^n$ in $\mathbb{C}P^n$, provides an intriguing avenue for further exploration.

Conclusion

The paper by Biran and Cornea represents a substantial contribution to the paper of Lagrangian submanifolds, combining intricate theoretical development with practical applications. It provides both a robust framework for understanding Lagrangian topology and motivates new investigations into the symplectic geometry's interplay with quantum invariants, fundamentally enriching the landscape of geometric topology.