##
**A constant rank theorem for Hermitian \(k\)-convex solutions of complex Laplace equations.**
*(English)*
Zbl 1192.58011

The main result of the paper establishes a constant rank theorem for the Hermitian \(k\)-convex solutions of the complex Laplace equation. The proof is based on the strong maximum principle.

In some geometry and analysis problems, it is always important to get the existence of convex solutions for partial differential equations. Up to now, we have in general two methods to produce convex solutions for elliptic partial differential equations. They are the macroscopic and microscopic convexity principle.

The macroscopic convexity principle was developed by Korevaar, Kennington, Kawohl, and by Alvarez-Lasry-Lions for generally nonlinear partial differential equations.

The microscopic convexity principle concentrates on the constant rank theorem for convex solutions of partial differential equations.

In differential geometry, there is much interest on the \(k\)-convexity of the solutions of the geometric partial differential equations. Here, the \(k\)-convexity means that the sum of any \(k\) eigenvalues of the Hessian matrix of the solution is nonnegative. Similarly, we can formulate the notions of \(k\)-convexity for the curvature operator and second fundamental forms of hypersurfaces.

For the potential application and independently interesting in complex analysis and complex geometry, in this paper, we establish the complex counterpart of the constant rank theorem on the Hermitian \(k\)-convex solutions for the complex Laplace equation.

We consider the following equation

\[ \Delta u=\sum^n_{\alpha=1}\frac{\partial^2u}{\partial z^\alpha\partial\overline z^\alpha}=f(z)\tag{1.1} \]

in a domain \(\Omega\) of \(\mathbb C^n\). Here, we assume that

\[ f\in C^{2,\lambda}(\Omega),\quad f>0,\text{ for some }0<\lambda <1.\tag{1.2} \]

We are interested in the solution \(u\) of Hermitian \(k\)-convex type. Namely, the sum of any \(k\) eigenvalues of the Hermitian matrix \((u_{\alpha\overline\beta})_{1\leq \alpha,\beta\leq n}\) is nonnegative. We would like to know, under what condition would the Hermitian \(k\)-convex solution \(u\) have a certain constant rank in \(\Omega\).

A Hermitian matrix \((H_{\alpha\overline\beta})\) is called \(k\)-concave, if the sum of any \(k\) eigenvalues of \((H_{\alpha\overline\beta})\) is nonpositive, equivalently, if \(\lambda_1+\cdots +\lambda_k < 0\), where \(\lambda_1\geq\lambda_2\geq\cdots\geq \lambda_n\) are the eigenvalues of \((H_{\alpha\overline\beta})\). In particular, a 1-concave Hermitian matrix is nonpositive definite. A \(C^2\)-function \(v\) is called Hermitian \(k\)-concave in \(\Omega\) if \((v_{\alpha\overline\beta})\) is \(k\)-concave in \(\Omega\).

Here is our main result:

Theorem 1.1. Let \(u\) and \(f\) satisfy (1.1), (1.2). Suppose that the \(w\)-matrix is positive semidefinite on \(\Omega\). Then the \(w\)-matrix has constant rank in \(\Omega\), if the Hermitian matrix

\[ (kff_{\alpha\overline\beta}-f_\alpha f_{\overline\beta})_{1\leq\alpha,\beta\leq n} \]

is \(k\)-concave in \(\Omega\). Equivalently, the \(w\)-matrix has constant rank if \(f^{1-1/k}\) is Hermitian \(k\)-concave for \(k\geq 2\), and \(-\log f\) is plurisubharmonic when \(k=1\).

In some geometry and analysis problems, it is always important to get the existence of convex solutions for partial differential equations. Up to now, we have in general two methods to produce convex solutions for elliptic partial differential equations. They are the macroscopic and microscopic convexity principle.

The macroscopic convexity principle was developed by Korevaar, Kennington, Kawohl, and by Alvarez-Lasry-Lions for generally nonlinear partial differential equations.

The microscopic convexity principle concentrates on the constant rank theorem for convex solutions of partial differential equations.

In differential geometry, there is much interest on the \(k\)-convexity of the solutions of the geometric partial differential equations. Here, the \(k\)-convexity means that the sum of any \(k\) eigenvalues of the Hessian matrix of the solution is nonnegative. Similarly, we can formulate the notions of \(k\)-convexity for the curvature operator and second fundamental forms of hypersurfaces.

For the potential application and independently interesting in complex analysis and complex geometry, in this paper, we establish the complex counterpart of the constant rank theorem on the Hermitian \(k\)-convex solutions for the complex Laplace equation.

We consider the following equation

\[ \Delta u=\sum^n_{\alpha=1}\frac{\partial^2u}{\partial z^\alpha\partial\overline z^\alpha}=f(z)\tag{1.1} \]

in a domain \(\Omega\) of \(\mathbb C^n\). Here, we assume that

\[ f\in C^{2,\lambda}(\Omega),\quad f>0,\text{ for some }0<\lambda <1.\tag{1.2} \]

We are interested in the solution \(u\) of Hermitian \(k\)-convex type. Namely, the sum of any \(k\) eigenvalues of the Hermitian matrix \((u_{\alpha\overline\beta})_{1\leq \alpha,\beta\leq n}\) is nonnegative. We would like to know, under what condition would the Hermitian \(k\)-convex solution \(u\) have a certain constant rank in \(\Omega\).

A Hermitian matrix \((H_{\alpha\overline\beta})\) is called \(k\)-concave, if the sum of any \(k\) eigenvalues of \((H_{\alpha\overline\beta})\) is nonpositive, equivalently, if \(\lambda_1+\cdots +\lambda_k < 0\), where \(\lambda_1\geq\lambda_2\geq\cdots\geq \lambda_n\) are the eigenvalues of \((H_{\alpha\overline\beta})\). In particular, a 1-concave Hermitian matrix is nonpositive definite. A \(C^2\)-function \(v\) is called Hermitian \(k\)-concave in \(\Omega\) if \((v_{\alpha\overline\beta})\) is \(k\)-concave in \(\Omega\).

Here is our main result:

Theorem 1.1. Let \(u\) and \(f\) satisfy (1.1), (1.2). Suppose that the \(w\)-matrix is positive semidefinite on \(\Omega\). Then the \(w\)-matrix has constant rank in \(\Omega\), if the Hermitian matrix

\[ (kff_{\alpha\overline\beta}-f_\alpha f_{\overline\beta})_{1\leq\alpha,\beta\leq n} \]

is \(k\)-concave in \(\Omega\). Equivalently, the \(w\)-matrix has constant rank if \(f^{1-1/k}\) is Hermitian \(k\)-concave for \(k\geq 2\), and \(-\log f\) is plurisubharmonic when \(k=1\).

Reviewer: Dumitru Motreanu (Perpignan)