Skip to main content
MathOverflow

Return to Question

fixed math display
Source Link
Willie Wong
Willie Wong
  • 39k
  • 4
  • 94
  • 176

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi, & -\partial_\bar{k}\phi ; \\ -\partial_j\phi, & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*}$$ \begin{pmatrix} -\phi & -\partial_\bar{k}\phi \\ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \end{pmatrix} $$ have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi, & -\partial_\bar{k}\phi ; \\ -\partial_j\phi, & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') $$ \begin{pmatrix} -\phi & -\partial_\bar{k}\phi \\ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \end{pmatrix} $$ have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

added 3 characters in body
Source Link
HKSHLZW
HKSHLZW
  • 399
  • 5
  • 14

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi & -\partial_\bar{k}\phi \\ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*}\begin{equation*} \left( \begin{array}{cc} -\phi, & -\partial_\bar{k}\phi ; \\ -\partial_j\phi, & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi & -\partial_\bar{k}\phi \\ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi, & -\partial_\bar{k}\phi ; \\ -\partial_j\phi, & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!

Source Link
HKSHLZW
HKSHLZW
  • 399
  • 5
  • 14

how to prove the relationship between pseudoconvexity and the monge-ampere matrix?

In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') \begin{equation*} \left( \begin{array}{cc} -\phi & -\partial_\bar{k}\phi \\ -\partial_j\phi & -\partial_{j\bar{k}}^2\phi \\ \end{array} \right) \end{equation*} have precise one negative eigenvalue and n positive eigenvalues .

So my question is how to prove this ? Anybody knows? Thanks very much!