20,414 research outputs found

    On estimates for the ∂ˉ\displaystyle \bar \partial equation in Stein manifolds

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    We generalize to intersection of strictly cc -convex domains in Stein manifold, Lr−Ls L^{r}-L^{s} and Lipschitz estimates for the solutions of the ∂ˉ \bar \partial equation done by Ma and Vassiliadou for domains in Cn. {\mathbb{C}}^{n}. For this we use a Docquier-Grauert holomorphic retraction plus the raising steps method I introduce earlier. This gives results in the case of domains with low regularity, C3, {\mathcal{C}}^{3}, for their boundary.Comment: I follow the nice suggestions done by the referee which substantially simplify the proof of theorem 4.1. Also typos are corrected and the presenrtation is slightly modifie

    Optimal Cobordisms between Torus Knots

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    We construct cobordisms of small genus between torus knots and use them to determine the cobordism distance between torus knots of small braid index. In fact, the cobordisms we construct arise as the intersection of a smooth algebraic curve in C2\mathbb{C}^2 with the unit 4-ball from which a 4-ball of smaller radius is removed. Connections to the realization problem of AnA_n-singularities on algebraic plane curves and the adjacency problem for plane curve singularities are discussed. To obstruct the existence of cobordisms, we use Ozsv\'ath, Stipsicz, and Szab\'o's Υ\Upsilon-invariant, which we provide explicitly for torus knots of braid index 3 and 4.Comment: 24 pages, 7 figures. Version 3: Minor corrections, implementation of referee's recommendations. Comments welcom
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