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.

Let G be a connected Lie group. We show that all characteristic classes of G are bounded—when viewed in the cohomology of the classifying space of the group G with the discrete topology—if and only if the derived group of the radical of G is simply connected in its Lie group topology. We also give equivalent conditions in terms of stable commutator length and distortion.

We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.

We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces$S$which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound$b$. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0.

Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.