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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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Families of Conformally Covariant Differential Operators, Q-Curvature and Holography (Hardcover, 2009 ed.)
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Families of Conformally Covariant Differential Operators, Q-Curvature and Holography (Hardcover, 2009 ed.)
Series: Progress in Mathematics, 275
Expected to ship within 10 - 15 working days
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A basic problem in geometry is to ?nd canonical metrics on smooth
manifolds. Such metrics can be speci?ed, for instance, by curvature
conditions or extremality properties, and are expected to contain
basic information on the topology of the underlying manifold.
Constant curvature metrics on surfaces are such canonical metrics.
Their distinguished role is emphasized by classical uniformization
theory. Amorerecentcharacterizationofthesemetrics describes them
ascriticalpoints of the determinant functional for the
Laplacian.The key tool here is Polyakov'sva- ationalformula for the
determinant. In higher dimensions, however,it is necessary to
further restrict the problem, for instance, to the search for
canonical metrics in conformal classes. Here two metrics are
considered to belong to the same conf- mal class if they di?er by a
nowhere vanishing factor. A typical question in that direction is
the Yamabe problem ([165]), which asks for constant scalar
curvature metrics in conformal classes. In connection with the
problem of understanding the structure of Polyakov type formulas
for the determinants of conformally covariant di?erential operators
in higher dimensions, Branson ([31]) discovered a remarkable
curvature quantity which now is called Branson's Q-curvature. It is
one of the main objects in this book.
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