53 research outputs found
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
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
Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces
We define a very general "parametric connect sum" construction which can be
used to eliminate isolated conical singularities of Riemannian manifolds. We
then show that various important analytic and elliptic estimates, formulated in
terms of weighted Sobolev spaces, can be obtained independently of the
parameters used in the construction. Specifically, we prove uniform estimates
related to (i) Sobolev Embedding Theorems, (ii) the invertibility of the
Laplace operator and (iii) Poincare' and Gagliardo-Nirenberg-Sobolev type
inequalities.
Our main tools are the well-known theories of weighted Sobolev spaces and
elliptic operators on "conifolds". We provide an overview of both, together
with an extension of the former to general Riemannian manifolds.
For a geometric application of our results we refer the reader to our paper
"Special Lagrangian conifolds, II: Gluing constructions in C^m".Comment: Minor changes, improved presentation. Final version. To appear in CA
Maslov, Chern-Weil and Mean Curvature
We provide an integral formula for the Maslov index of a pair over a
surface , where is a complex vector bundle and
is a totally real subbundle. As in Chern-Weil
theory, this formula is written in terms of the curvature of plus a
boundary contribution.
When is obtained via an immersion of into a
pair where is K\"ahler and is totally real, the formula allows
us to control the Maslov index in terms of the geometry of . We exhibit
natural conditions on 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
From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness
Given a minimal Lagrangian submanifold L in a negative Kaehler--Einstein
manifold M, we show that any small Kaehler--Einstein perturbation of M induces
a deformation of L which is minimal Lagrangian with respect to the new
structure. This provides a new source of examples of minimal Lagrangians. More
generally, the same is true for the larger class of totally real J-minimal
submanifolds in Kaehler manifolds with negative definite Ricci curvature.Comment: Final version, 22 pages; to appear in a special volume in memory of
Paolo de Bartolomeis, Boll. UM
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