Conformal deformations of the smallest eigenvalue of the Ricci tensor
Pengfei Guan and Guofang Wang
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Submission date: 05. May. 2005
published in: American journal of mathematics, 129 (2007) 2, p. 499-526
DOI number (of the published article): 10.1353/ajm.2007.0011
MSC-Numbers: 53C21, 35J6, 58E11
Keywords and phrases: conformal deformation, ricci tensor, minimal volume, fully nonlinear equation, local estimates
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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.