Conformal deformations of the smallest eigenvalue of the Ricci tensor

Details Conformal deformations of the smallest eigenvalue of the Ricci tensor Conformal deformations of the smallest eigenvalue of the Ricci tensor

2005-05-05

arxiv.org/abs/math/0505083v1

Mathematics

Differential Geometry

23 pages

Scientific paper

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the theory of fully nonlinear equations, we establish the existence of solutions for this equation.

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