Riemann Surfaces 1
Something Like Picard for 1-Forms
Picard’s great theorem is a statement about how wildly a holomorphic function can behave near an essential singularity. The conjecture below asks whether injectivity of local primitives of a 1-form is enough to rule out such wild behaviour at the origin, forcing the 1-form to extend meromorphically across the puncture. Conjecture (Elsner, 2010) Let $D$ be the open unit disk and let $U_1,\dots,U_n$ be open sets with $\bigcup_{j=1}^n U_j = D\setminus{0}$. Suppose there are injective holomorphic functions $f_j : U_j \to \mathbb{C}$ such that $$\mathrm{d}f_j = \mathrm{d}f_k \quad \text{on every connected component of } U_j \cap U_k.$$ Then the $\mathrm{d}f_j$ glue together to a meromorphic 1-form on $D$.