Special Lagrangian conifolds, II: Gluing constructions in C^m

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity but are not, globally, cones. We also obtain: (i) a desingularization procedure for transverse intersection and self-intersection points, using "Lawlor necks"; (ii) a construction which completely desingularizes any SL conifold by replacing isolated conical singularities with non-compact asymptotically conical (AC) ends; (iii) a proof that there is no upper bound on the number of AC ends of a SL conifold; (iv) the possibility of replacing a given collection of conical singularities with a completely different collection of conical singularities and of AC ends. As a corollary of (i) we improve a result by Arezzo and Pacard concerning minimal desingularizations of certain configurations of SL planes in C^m, intersecting transversally.Comment: Several new results. Final version. To appear in Proc. LM

Special Lagrangian conifolds, I: Moduli spaces

We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends. Our main theorem is the natural next step in the chain of results initiated by McLean and continued by the author and by Joyce. We emphasize a unifying framework for studying the various cases and discuss analogies and differences between them. This paper also lays down the geometric foundations for our paper "Special Lagrangian conifolds, II" concerning gluing constructions for SL conifolds in C^m.Comment: This is the final version, to appear in Proc. LMS. I have also posted on arXiv an "extended version" of this paper, including many additional details of possible interes

Maslov, Chern-Weil and Mean Curvature

We provide an integral formula for the Maslov index of a pair $(E,F)$ over a surface $\Sigma$, where $E\rightarrow\Sigma$ is a complex vector bundle and $F\subset E_{|\partial\Sigma}$ is a totally real subbundle. As in Chern-Weil theory, this formula is written in terms of the curvature of $E$ plus a boundary contribution. When $(E,F)$ is obtained via an immersion of $(\Sigma,\partial\Sigma)$ into a pair $(M,L)$ where $M$ is K\"ahler and $L$ is totally real, the formula allows us to control the Maslov index in terms of the geometry of $(M,L)$. We exhibit natural conditions on $(M,L)$ which lead to bounds and monotonicity results.Comment: v3: same results, 11 pages, final version. To appear in Journal of Geometry and Physic

Extremal length in higher dimensions and complex systolic inequalities

Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole.Comment: Contains improved presentation of complex systolic inequalities and comparisons with special Lagrangian geometry. Various other minor changes. To appear in Journal of Geometric Analysi