8,322 research outputs found
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
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
Mean Curvature Flow, Orbits, Moment Maps
Given a compact Riemannian manifold together with a group of isometries, we
discuss MCF of the orbits and some applications: eg, finding minimal orbits. We
then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in
the Kaehler-Einstein case we find a relation between MCF and moment maps which,
for example, proves that the minimal Lagrangian orbits are isolated.Comment: 18 pages; minor change
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