Calibrations

This web page contains material for the workshop Calibrations.

Preliminary list of open problems: dvi, postscript or pdf.

Suggested reading for Dominic Joyce's talk on the first day: Singularities of special Lagrangian submanifolds.

A few references:

Selman Akbulut, Sema Salur:
Calibrated Manifolds and Gauge Theory,
http://arxiv.org/abs/math.GT/0402368
Remark: This paper discusses relation between calibrations and
Seiberg-Witten equations.

Mark Gross:
Special Lagrangian Fibrations I & II
http://arxiv.org/abs/alg-geom/9710006, math.AG/9809072
Remark: These papers studies topological and geometrical
properties of special lagrangian fibrations in Calabi-Yau manifolds,
assuming such fibrations exist.

Harvey, R. and Lawson, B.:
Plurisubharmonic Functions in Calibrated Geometries
http://arxiv.org/abs/math.CV/0601484
Remark: This paper studies the notion of plurisubharmonic functions in
calibrated geometry.
These functions generalize the classical plurisubharmonic functions from
complex geometry and enjoy
many of their important properties.

Dominic Joyce:
Singularities of special Lagrangian submanifolds
http://arxiv.org/abs/math.DG/0310460
Remark: This is a survey on what is known about singularities of special
Lagrangian submanifolds
in Calabi-Yau manifolds. The bulk of the paper summarizes Joyce's five
papers
math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356,
math.DG/0303272
on SL m-folds X with isolated conical singularities.

Naichung Conan Leung
Riemannian geometry over differential normed division algebra,
http://arxiv.org/abs/math.DG/0303153
Remark: This paper generalize SYZ conjecture for mirror symmetry to other
manifolds with exceptional holonomy groups.

Wei-Dong Ruan:
Generalized special Lagrangian torus fibration for Calabi-Yau
hypersurfaces in toric varieties
I, II, III (http://arxiv.org/abs/math.DG/0303114, 0303278, 0309450)
Remark: These papers study generalized special Lagrangian torus fibrations
for Calabi-Yau hypersurfaces in toric variety near the large complex
limit, with respect to the
restriction of a toric metric on the toric variety to the Calabi-Yau
hypersurface.

Andrew Strominger, Shing-Tung Yau, Eric Zaslow:
Mirror symmetry is $T$-duality,
http://arxiv.org/abs/hep-th/9606040
Remark: The SYZ conjecture for constructing mirrors of Calabi-Yau
manifolds through special Lagrangian tori is posted in this paper.

Gang Tian:
Gauge geometry and calibrated geometry,
http://arxiv.org/abs/math.DG/0010015
Remark: This paper studied anti-self-dual connections defined by
codimension-4 calibrations in
manifolds of arbitrary dimensions.

Yu Yuan:
Global solutions to special Lagrangian equations
http://arxiv.org/abs/math.AP/0501456
Remark: This paper shows that any global solution to the special
Lagrangian equations with
the phase larger than a critical value must be quadratic


The following papers construct examples of special Lagrangian submanifolds
and (co)associated submanifolds:

Robert L. Bryant:
Some examples of special Lagrangian tori
http://arxiv.org/abs/math.DG/9902076

Emma Carberry, Ian McIntosh:
Minimal Lagrangian 2-tori in $\mb{CP}^2$ come in real families of every
dimension,
http://arxiv.org/abs/math.DG/0308031

M. Haskins, N. Kapouleas:
Special Lagrangian cones with higher genus links,
http://arxiv.org/abs/math.DG/0512178

Shengli Kong, Erxiao Wang, Chuu-Lian Terng:
Associative Cones and Integrable Systems,
http://arxiv.org/abs/math.DG/0602565

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A list of participants is available.