We state and prove a long-elusive relation between genus-one Gromov-Witten of a complete intersection and twisted Gromov Witten invariants of the ambient projective space. As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component M 0 1,k(Pn, d) of the moduli space of stable genus-one holomorphic maps into Pn have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, J-holomorphic maps. This extension is used to show that these cones are the correct genus-one analogues of the vector bundles relating genuszero Gromov-Witten invariants of a complete intersection to those of the ambient projective space. A relationship for higher-genus invariants is conjectured as well.

We discuss a method for deriving differential equations for the prepotential and the mirror map arising from a pair of Calabi-Yau manifold. Examples with one Kahler modulus are given. Here we nd that the differential equations we derive almost entirely characterize the prepotential and the mirror map in question. Some of the results here have been announced previously in

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.

We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordstr\"{o}m manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space. In particular, Corollary 1.8 implies that the embeddedness condition in Theorem 2 of [Brendle-Eichmair, JDG. 2013] is not necessary.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques-Neves in their proof of the Willmore conjecture \cite{MN12}, is a bridge to connect sweepouts appearing naturally in geometry to sweepouts used in the min-max theory. Our result removes a critical assumption of \cite{MN12}, called the no mass concentration condition, and hence confirms a conjecture by Marques-Neves in \cite{MN12}.