169 research outputs found
Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
We present a local classification of conformally equivalent but oppositely
oriented 4-dimensional Kaehler metrics which are toric with respect to a common
2-torus action. In the generic case, these "ambitoric" structures have an
intriguing local geometry depending on a quadratic polynomial q and arbitrary
functions A and B of one variable.
We use this description to classify Einstein 4-metrics which are hermitian
with respect to both orientations, as well a class of solutions to the
Einstein-Maxwell equations including riemannian analogues of the
Plebanski-Demianski metrics. Our classification can be viewed as a riemannian
analogue of a result in 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.
These Einstein metrics are precisely the ambitoric structures with vanishing
Bach tensor, and thus have the property that the associated toric Kaehler
metrics are extremal (in the sense of E. Calabi). Our main results also
classify the latter, providing new examples of explicit extremal Kaehler
metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A
and B are quartic polynomials, but with different conditions on the
coefficients. In the sequel to this paper we consider global examples, and use
them to resolve the existence problem for extremal Kaehler metrics on toric
4-orbifolds with second betti number b2=2.Comment: 31 pages, 1 figure, partially replaces arXiv:1010.099
Refined Kato inequalities and conformal weights in Riemannian geometry
We establish refinements of the classical Kato inequality for sections of a
vector bundle which lie in the kernel of a natural injectively elliptic
first-order linear differential operator. Our main result is a general
expression which gives the value of the constants appearing in the refined
inequalities. These constants are shown to be optimal and are computed
explicitly in most practical cases.Comment: AMS-LaTeX, 36pp, 1 figure (region.eps
Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
We provide an explicit resolution of the existence problem for extremal
Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More
precisely we show that M admits such a metric if and only if its rational
Delzant polytope (which is a labelled quadrilateral) is K-polystable in the
relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi
et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric,
i.e., compatible with a conformally equivalent, oppositely oriented toric
Kaehler metric, which turns out to be extremal as well. These results provide a
computational test for the K-stability of labelled quadrilaterals.
Extremal ambitoric structures were classified locally in Part I of this work,
but herein we only use the straightforward fact that explicit Kaehler metrics
obtained there are extremal, and the identification of Bach-flat (conformally
Einstein) examples among them. Using our global results, the latter yield
countably infinite families of compact toric Bach-flat Kaehler orbifolds,
including examples which are globally conformally Einstein, and examples which
are conformal to complete smooth Einstein metrics on an open subset, thus
extending the work of many authors.Comment: 31 pages, 3 figures, partially replaces and extends arXiv:1010.099
Cheating and the evolutionary stability of mutualisms
Interspecific mutualisms have been playing a central role in the functioning of all ecosystems since the early history of life. Yet the theory of coevolution of mutualists is virtually nonexistent, by contrast with well-developed coevolutionary theories of competition, predator–prey and host–parasite interactions. This has prevented resolution of a basic puzzle posed by mutualisms: their persistence in spite of apparent evolutionary instability. The selective advantage of 'cheating', that is, reaping mutualistic benefits while providing fewer commodities to the partner species, is commonly believed to erode a mutualistic interaction, leading to its dissolution or reciprocal extinction. However, recent empirical findings indicate that stable associations of mutualists and cheaters have existed over long evolutionary periods. Here, we show that asymmetrical competition within species for the commodities offered by mutualistic partners provides a simple and testable ecological mechanism that can account for the long-term persistence of mutualisms. Cheating, in effect, establishes a background against which better mutualists can display any competitive superiority. This can lead to the coexistence and divergence of mutualist and cheater phenotypes, as well as to the coexistence of ecologically similar, but unrelated mutualists and cheaters
Potential one-forms for hyperk\"ahler structures with torsion
It is shown that an HKT-space with closed parallel potential 1-form has
-symmetry. Every locally conformally hyperk\"ahler manifold
generates this type of geometry. The HKT-spaces with closed parallel potential
1-form arising in this way are characterized by their symmetries and an
inhomogeneous cubic condition on their torsion.Comment: 16 pages, Latex, no figure
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