Translation surface ($(X,\omega )$)

A surface with an atlas outside finitely many cone points whose changes of coordinate are translations. On a compact surface it is equivalent to a nonzero holomorphic one-form. A zero of order $m$ has cone angle $2\pi (m+1)$.

Cut equal straight slits in two translation surfaces of genus one and cross-glue their banks by translation. The connected sum of oriented manifolds has genus two and two cone points of angle $4\pi$, hence two simple zeros of its holomorphic one-form. Choosing one torus of side $\delta$ and a slit of length ${\delta }^2$ creates a flat-small handle without forcing its generator to have small extremal length.