起先我們觀察賀深矩陣與擬凸之間的密切關係, 如果賀深矩陣是正定的那麼我可以確定這個域會是擬凸. 所以我們可以確定這個域是擬凸, 只要觀察賀深矩陣的固有值是否都是正的, 最後我們用超曲面的標準型來判定一個域是否為擬凸 並且我給了兩個例子計算驗證此定理.

We will discuss the relationship between complex Hessian matrices and the Levi forms.
If the complex Hessian matrix is positive definite, the associated Levi form must be positive. Thus, we can
ensure the Levi pseudoconvexity by observing the eigenvalues of the respected complex Hessian matrix.
Finally, we will use the Chern-Moser normal form to look up the real analytic defining function of the domain
$D$ in $\mathbb{C}^{n+1}$.
verify pseudoconvexity of a domain at a boundary point. At the end a couple of working examples will be given.

1. Diederich, K., Forn\ae ss, J.E., Pseudoconvex
domains: An Example with Nontrivial Nebenhulle
Math. Ann (1977), 275--292.
2. Diederich, K., Forn\ae ss, J.E., Pseudoconvex domains: Bounded Strictly Plurisubharmonic Exhaustion Functions
Invent. Math. 39 (1977), no. 2, 129--141.
3.Moser, J.K., Webster S.M.,: Normal forms for real surface in C^2 near complex tangents and
hyperbolic surface transformation}}, Acta Math. 150 (1983), no. 3--4, 255--296.
4. Chern, S.S., Moser J.K.: Real hypersurfaces in complex manifolds Acta Math. 150 (1983), no. 3--4, 219--271.
5. Chen, S.-C., Shaw, M.-C., Partial Differential Equations in Several Complex Variable, AMS/IP 19.