Biholomorphically equivalent domains

Definition

Let $U,V\subset \mathbb {C} ^{n}$ be domains (open connected subset). We say that $U,V$ are biholomorphically equivalent if there exists a holomorphic function $\varphi :U\to V$ with a holomorphic inverse $\varphi ^{-1}:V\to U$.

(By a holomorphic function from $U$ to $V$, we mean a holomorphic function from $U$ to $\mathbb {C} ^{n}$, whose image lies completely inside $V$. In this case, we get a bijection from $U$ to $V$ that is holomorphic both ways.

When $n=1$, biholomorphically equivalent domains are also termed conformally equivalent.

Relation with other equivalence relations

Weaker equivalence relations

• Diffeomorphic domains
• Homeomorphic domains
• Homotopy-equivalent domains

Facts

Any two simply connected open subsets of $\mathbb {C}$ are biholomorphically equivalent. This is a consequence of the Riemann mapping theorem.