The Eigenvalue Problem for the Complex Hessian Operator on m-Pseudoconvex Manifolds

We establish C^1,1-regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly m-pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue.