Heegaard Floer correction terms and rational genus bounds (Q462291)

For a rationally null-homologous knot \(K\) in a closed oriented \(3\)-manifold \(Y\), the rational genus of \(K\) was introduced by \textit{D. Calegari} and \textit{C. Gordon} [Comment. Math. Helv. 88, No. 1, 85--130 (2013; Zbl 1275.57019)] by using a rational Seifert surface. Moreover, \textit{V. Turaev} [Algebr. Geom. Topol. 7, 135--156 (2007; Zbl 1145.57011)] defined a function on torsion classes of \(H_1(Y)\) from rational genera. When the 1st Betti number of \(Y\) is positive, there is a known lower bound for Turaev's function. The main result of the paper under review gives such a lower bound for a rational homology \(3\)-sphere \(Y\) in terms of the correction terms in Heegaard Floer homology. In particular, if \(Y\) is an \(L\)-space, then the above evaluation implies that a Floer simple knot is genus minimizing in its homology class. This solves affirmatively Rasmussen's conjecture which claims that simple knots in lens spaces are genus minimizing.