Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds

Details Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds

2010-10-05

arxiv.org/abs/1010.0992v1

Mathematics

Differential Geometry

43 pages

Scientific paper

We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to either orientation. This can be considered as a riemannian analogue of a result in General Relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. We discuss toric compactifications of these metrics on orbifolds and provide infinite discrete families of compact toric extremal Kaehler orbifolds. Our examples include Bach-flat Kaehler orbifolds which are conformal to complete smooth Einstein metrics on an open subset. We illustrate how these examples fit with recent conjectures relating the existence of extremal toric metrics to various notions of stability.

Affiliated with

Also associated with

No associations

Rating

Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-87610