Abstract
Generalised Einstein equations (Einstein equations with sources in thephysicist's grammar) can, in the Kähler setup, be seen ascohomological equations within the first Chern class. Introducing a twoparameter secondary class (or source term) to prescribe such acohomological relation, we characterise regions and paths of thoseparameters to ensure that the associated equation admits at least onesolution. Those regions could be seen, in the context of geometricanalysis, as a measure of the metric flexibility allowed within theKählerian rigidity. When the first Chern class is positive, the AubinTian constant and the bounds for the pluriharmonic concavity andconvexity of the source term characterise the bounds of that region.Taking into account the minimal regularity of the secondary class toensure the existence of classical solutions, we observe, in particular,an improvement of some results quoted in the literature in the contextof Calabi's conjecture.
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Pons, D. Generalised Einstein Equations and Prescribed Relations for the First Chern Weil Form. Annals of Global Analysis and Geometry 25, 177–200 (2004). https://doi.org/10.1023/B:AGAG.0000018557.28153.ff
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DOI: https://doi.org/10.1023/B:AGAG.0000018557.28153.ff