Rigidity for general semiconvex entire solutions to the sigma-2 equation
The equation \[ \sigma_2(D^2 u)\equiv \sum\limits_{1\le i<j\le n} \lambda_i\lambda_j= 1, \] where \(\lambda_1,\lambda_2,\dots, \lambda_n\) are the eigenvalues of the Hessian matrix \[ D^2u= \left(\frac{\partial^2 u}{\partial x_i\partial x_j}\right)_{m\times n} \] has many applications. In two variables it reduces to the Monge-Ampère equation \[ u_{xx} u_{yy}- u^2_{xy}= 1. \] The contribution of this work is a proof of the theorem below, which was conjectured in 2009 by one of the authors. Theorem. Let \(u= u(x_1,x_2,\dots,x_n)\) be a smooth solution of the equation \(\sigma_2(D^2u)= 1\) in the whole \(\mathbb{R}^n\). If \(D^2u\ge -KI\) for a constant \(K\), then \(u\) is a quadratic polynomial. For \(n=2\) this is Jörgen's classical theorem for the Monge-Ampère equation. The case \(n=3\) has been proved earlier by Y. Yuan. The present proof is technically advanced, based on PDE methods and identities from differential geometry. There are non-polynomial entired solutions that do not satisfy the quasi-convexity assumptions \(D^2u\ge -KI\).
scientific article; zbMATH DE number 7608261
scientific article