The geometric complexity of special Lagrangian T2-cones

By M. Haskins. 2003

68 citations recorded by the source corpus.

We prove a number of results on the geometric complexity of special Lagrangian (SLG) T2-cones in ℂ3. Every SLG T2-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T2-cone gives a lower bound for its geometric complexity, i.e. the area, the stability index and the Legendrian index of any SLG T2-cone are all bounded below by explicit linearly growing functions of the spectral curve genus. We prove that the cone on the Clifford torus (which has spectral curve genus zero) in S5 is the unique SLG T2-cone with the smallest possible Legendrian index and hence that it is the unique stable SLG T2-cone. This leads to a classification of all rigid “index 1” SLG cone types in dimension three. For cones with spectral curve genus two we give refined lower bounds for the area, the Legendrian index and the stability index. One consequence of these bounds is that there exist S1-invariant SLG torus cones of arbitrarily large area, Legendrian and stability indices. We explain some consequences of our results for the programme (due to Joyce) to understand the “most common” three-dimensional isolated singularities of generic families of SLG submanifolds in almost Calabi-Yau manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/740537cc374d01a1f7f58ccdf5720a6e017980ee

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In Applied Mathematics

1. Iskander A. Taimanov
2. Haizhong Li
3. Dmitry Berdinsky
4. I. Castro
5. F. Urbano
6. Ben Andrews

In Geometry and Topology

1. Iskander A. Taimanov
2. Haizhong Li
3. Dmitry Berdinsky
4. I. Castro
5. F. Urbano
6. Ben Andrews

Citations in the corpus

• Surfaces in Three-Dimensional Lie Groups (2005, methodology): “in their conformal classes as the final step. Recently M. Haskins proposed to treat the genus of the spectral curve as a measure of geometric complexity for some other variational problems of geometry [6]. In the present paper we derive the analogous formulas for the functional E(M) for surfaces in N”
• HAMILTONIAN-MINIMAL LAGRANGIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS (2004, methodology): “ manifold Mand C(φ) : R× M→ Cn+1 the cone with link φ. Then φis C-minimal if and only if C(φ) is H-minimal. In particular, φis minimal if and only if C(φ) is minimal; this result was used in [H1] and [H2]. Thanks to Proposition 3 we have a fruitful simple method of construction of examples of H-mini”