Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems

Abstract

We introduce a new class of curvature PDOs describing relevant properties of real hypersurfaces of C nþ1 : In our setting, the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second-order fully nonlinear PDOs not elliptic at any point. However, when computed on generalized s-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures. r

Levi Curvature with Radial Symmetry: a Sphere Theorem for Bounded Reinhardt Domains of $\mathbb C^2$

“…The domain D is strictly Levi-pseudoconvex iff the Levi curvature is strictly positive at any point of @D. The Levi curvature is independent of the defining function F (see, e.g., [5]): nevertheless, it implicitly depends on the orientation of @D. Moreover, by direct computation we obtain that…”

“…To this end, we will use the following crucial result proved in [5]. Firstly, we show that f (0) determines f H (0) and f HH (0).…”

“…In analogy with the real case, in the case of C n1 one defines K (j) to be the Levi curvature of order j (1 j n) by taking the jth elementary symmetric function of the eigenvalues of the Levi form: for details we refer to [5]. The Levi curvature of order n (the product of the n eigenvalues) is called the total Levi curvature: in the case n 1 it coincides with the definition introduced above.…”

Cauchy–Riemann meet Monge–Ampère

“…, λ n−1 and define the mth complex mean curvature K m similarly as H m , so that K = K n−1 is the Levi curvature of the boundary. See [92] and [90] Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.…”

Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

“…However, in a joint paper with Lanconelli [12] we wrote the j-th Levi curvature for real hypersurfaces in C n+1 in terms of elementary symmetric functions of the eigenvalues of the normalized Levi form, and we proved a strong comparison principle, leading to symmetry theorems for domains with constant curvatures.…”