Mathematics – Differential Geometry
Scientific paper
2008-04-29
Mathematics
Differential Geometry
19 pages, title and abstract changed, major revisions/additions, particularly Section 3
Scientific paper
A holomorphy potential is a complex valued function whose complex gradient, with respect to some K\"ahler metric, is a holomorphic vector field. Given $k$ holomorphic vector fields on a compact complex manifold, form, for a given K\"ahler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where $k=0$ and the functional is the square of the $L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the \sk metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.
No associations
LandOfFree
Scalar curvature and holomorphy potentials 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 Scalar curvature and holomorphy potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Scalar curvature and holomorphy potentials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-210293