A sharp estimate for the genus of embedded surfaces in the 3-sphere (Q6564130)

Let \(\Sigma\) be a closed orientable surface embedded in a closed orientable Riemannian manifold \(M\) of dimension \(3\).\N\NIn this paper, starting from a hypothesis on the sectional curvature of the manifold \(M\), the author establishes a new estimate for the genus of the surface embedded in the manifold \(M\) as a function of an integral of the traceless second fundamental form of \(\Sigma\).\N\NThe author shows that this new estimate for the genus cannot be derived from the classical estimate involving the area and the \(L^2\) norm of the traceless second fundamental form of \(\Sigma\).