【摘要】
In this talk we report a recent joint work with Biao Ma from BICMR and ECNU.
The 1959 works of Gårding on hyperbolic polynomials and their associated convex cones has been heavily used in geometric analysis, in the context of Caffarelli-Nirenberg-Spruck's (CNS) theory for fully nonlinear elliptic PDE of Hessian type.
Generalizing earlier works of C.-M. Lin on $Upsilon$-stableness and some of our previous results, we introduce a class of multi-affine polynomials that includes key hyperbolic examples. A Gårding polynomial is characterized by properties of its associated Gårding cone and some important subclass may be directly used to construct new geometric PDEs within the CNS setting. We explore its relation to the Lorenzian polynomial theory of Brändén-Huh, which provides key local properties. We also discuss the construction of Gårding polynomials and several other aspects of our works. Finally we discuss some application in complex geometry, and pose several potential directions.
