complex Hessian matrixComplexHessianMatrix2004-08-02 21:12:502006-06-21 11:38:24Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereSuppose that $f \colon {\mathbb{C}}^n \to \mathbb{C}$ be twice differentiable
and let
\begin{equation*}
\frac{\partial}{\partial z_k} :=
\frac{1}{2}\left(
\frac{\partial}{\partial x_k} - i \frac{\partial}{\partial y_k}
\right)
\quad \text{ and } \quad
\frac{\partial}{\partial \bar{z}_k} :=
\frac{1}{2}\left(
\frac{\partial}{\partial x_k} + i \frac{\partial}{\partial y_k}
\right) .
\end{equation*}
Then the {\em \PMlinkescapetext{complex Hessian}} is the matrix
\begin{equation*}
\begin{bmatrix}
\frac{\partial^2 f}{\partial z_1 \partial \bar{z}_1} &
\frac{\partial^2 f}{\partial z_1 \partial \bar{z}_2} &
\ldots &
\frac{\partial^2 f}{\partial z_1 \partial \bar{z}_n}
\\
\frac{\partial^2 f}{\partial z_2 \partial \bar{z}_1} &
\frac{\partial^2 f}{\partial z_2 \partial \bar{z}_2} &
\ldots &
\frac{\partial^2 f}{\partial z_2 \partial \bar{z}_n}
\\
\vdots &
\vdots &
\ddots &
\vdots
\\
\frac{\partial^2 f}{\partial z_n \partial \bar{z}_1} &
\frac{\partial^2 f}{\partial z_n \partial \bar{z}_2} &
\ldots &
\frac{\partial^2 f}{\partial z_n \partial \bar{z}_n}
\end{bmatrix}
.
\end{equation*}
When applied to tangent vectors of the zero set of $f$,
it is called the Levi form and used to define a Levi
pseudoconvex point of a boundary of a domain. Note that the \PMlinkescapetext{complex Hessian}
matrix is not the same as the \PMlinkescapetext{normal} (real) Hessian. A twice continuously
differentiable real valued
function with a
positive semidefinite real Hessian matrix at every point is convex, but a function with
positive semidefinite \PMlinkescapetext{complex Hessian} matrix at every point is
plurisubharmonic (since it's
continuous it's also called a pseudoconvex function).
\begin{thebibliography}{9}
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}