On convex to pseudoconvex mappings

Theorem 1. Let F : U ′ → U be a C 2 -diffeomorphism. Then the following conditions are equivalent:

i) For every convex hypersurface M ′ ⊂ U ′ the image M = F (M ′ ) is a pseudoconvex hypersurface in U.

ii) The inverse map Φ := F -1 : U → U ′ satisfies the following second order PDE System ∂∂Φ = (dΦ -1 (∆Φ), dz) ∧ ∂Φ + (dz, dΦ -1 (∆Φ)) ∧ ∂Φ.

(1.1)

iii) The equation (1.1) has the following geometric meaning: for every z ∈ U and every

Here we use the following notation: for a vector v = (v 1 , …, v n ) ∈ C n and dz = (dz 1 , …, dz n ) we set (dz, v) = vj dz j and (v, dz) = v j dz j . Throughout this note we shall use the Einstein summation convention.

Remark 1. Pluriharmonic Φ-s clearly satisfy (1.1) (or (1.2)) and let us remark that this geometric characterization of pluriharmonic diffeomorphisms perfectly agrees with an analytic one: The class P of pluriharmonic diffeomorphisms C n → C n is stable under biholomorphic parametrization of the source and R-linear of the target. Really, these parametrization preserve accordingly pseudoconvexity and convexity of hypersurfaces. 2. The item (i) of the Theorem is clearly equivalent to the following one: For every strictly convex quadric

I.e, it is enough to check this condition only for quadrics. 3. The fact that (1.1) admits other solutions then just pluriharmonic mappings is very easy to see from the form of its linearization at identity ∂∂Φ = (∆Φ, dz) ∧ dz.

(1.3)

Remark that any map of the form Φ(z) = (ϕ 1 (z 1 ), …, ϕ n (z n )) satisfies (1.3) provided all ϕ j , except for some j 0 , are harmonic. And this ϕ j 0 can be then an arbitrary C 2 -function.

Denote by ζ = ξ + iη a tangent vector at point z ∈ C n . Recall that the real Hessian of a real valued function

(2.1)

A hypersurface M = {z ∈ U : ρ(z) = 0}, with ρ is C 2 -regular, ρ(0) = 0 and ∇ρ| M = 0, is strictly convex if the defining function ρ can be chosen with positive definite Hessian, i.e., H R ρ(z) (ζ, ζ) > 0 for all z ∈ M and all ζ = 0. One readily checks the following expression of the real Hessian of ρ in complex coordinates

Recall that the Hermitian part L ρ(z) (ζ, ζ) = ∂ 2 ρ ∂z i ∂ zj ζ i ζj of the Hessian is called the Levi form of ρ (and of M). M is strictly pseudoconvex if its Levi form is positive definite on the complex tangent space T c z M = {ζ ∈ T z C n : ∂ρ(z), ζ = 0} for every z ∈ M. Here (•, •) stands for the standard Hermitian scalar product in C n .

Let

Lemma 2.1. The Levi form of ρ at point z decomposes as

where

(2.5)

Proof. Here we denote by dΦ z is the differential of the inverse map Φ :

Denote by ν the vector with components ν j = ∂z ′ j ∂zα ζ α and by µ with

and

We need to get more information about the structure of both terms L 0 ρ and L 1 ρ of the Levi form. Let’s prove that the following relation holds

(2.9)

To see this we make the following change in (2.9):

We used the obvious relations

and the complex expression of the real Hessian (2.2). Therefore

(2.11) From (2.10) and (2.11) we get the formula (2.4) of the Lemma.

Remark 2. If the real Hessian of ρ ′ at z ′ is positive (resp. non-negative) definite then the component L 0 ρ(z) (ν, µ) of the Levi form of ρ at z = F (z ′ ) is also positive (resp. nonnegative) definite for any C 2 -germ of a diffeomorphism F . Now we turn to L 1 ρ . Note that in complex notations ∇ρ = ∂ρ as well as that standard Euclidean scalar product •, • in C n is equal to the real part of the Hermitian one (•, •). Therefore from (2.8) we get

) , which proves (2.5).

We start with the proof of the geometric characterization of convex to pseudoconvex mappings given in (iii) of the Theorem. By a complex (real) line in C n we mean an 1dimensional complex (real) subspace of C n . The same for complex (real) plain. Take a complex line l = span {ζ} in T z C n and let Π ′ ⊂ T z ′ C n be the real plain -image of l under dΦ z , i.e., Π ′ = span {dΦ z (ζ), dΦ z (iζ)}. Let l ′ := ∂∂Φ z (l) denotes the real (!) line -image of l under the mapping

We consider l ′ as a real line in T z ′ C n .

Lemma 3.1. Suppose that given a diffeomorphism F : U ′ → U. Then F sends convex quadrics to pseudoconvex hypersurfaces if and only if for every z ∈ U and for all l ′ := ∂∂Φ z (l) and Π ′ = dΦ z (l) as above one has l ′ ⊂ Π ′ .

Proof. Let us prove the “only if " assertion first. We may suppose that z ′ = 0 ′ . Take any strictly convex M ′ = {ρ ′ (z ′ ) = 0} defined by a C 2 -function ρ ′ with positive defined Hessian such that T 0 ′ M ′ ⊃ Π ′ . By M denote the image

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