Holomorphic mappings of once-holed tori

On sets of marked once-holed tori allowing holomorphic mappings into Riemann surfaces with marked handle

“…We prove Theorem 1 in §3 after summarizing results of [10] in §2. In §4 we compare T a [Y 0 ] with the set T σ [Y 0 ] of marked once-holed tori having longer geodesics than corresponding geodesics on Y 0 .…”

“…conformal) mapping of X into Y 0 . In our previous work [10] we introduced handle conditions to investigated geometric properties of T a [Y 0 ] and T c [Y 0 ]. We recall the definition of a handle condition.…”

“…Note, however, that Proposition 2 asserts that the identity H[P a ] = H(1/λ[P a ]) actually holds. On the other hand, in general,H(1/λ[P c ]) is a proper subset of H[P c ] (see [10,Example 13]). For the proof of Theorem 1 we define the handle covering surface of a Riemann surface Y 0 with marked handle.…”

“…The space T of marked once-holed tori is a three-dimensional real-analytic manifold with boundary. In our previous work [10], for a given Riemann surface Y 0 with marked handle, we investigated the set T a [Y 0 ] of marked once-holed tori X such that there is a holomorphic mapping of X into Y 0 . We introduced a new condition called a handle condition to obtain the following two results.…”

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In our previous work [10], for a given Riemann surface Y 0 with marked handle, we investigated geometric properties of the set of marked onceholed tori X allowing holomorphic mappings of X into Y 0 . It turned out that it is a closed domain with Lipschitz boundary.

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