Preprint 43/2005

Conformal deformations of the smallest eigenvalue of the Ricci tensor

Pengfei Guan and Guofang Wang

Contact the author: Please use for correspondence this email.
Submission date: 05. May. 2005
Pages: 25
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
Download full preprint: PDF (309 kB), PS ziped (263 kB)

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.

18.10.2019, 02:12