Based on earlier work of the latter two named authors on the higher super-Teichmueller space with N=1, a component of the flat OSp(1\2) connections on a punctured surface, here we extend to the case N=2 of flat OSp(2\2) connections. Indeed, we construct here coordinates on the higher super-Teichmueller space of a surface F with at least one puncture associated to the supergroup OSp(2\2), which in particular specializes to give another treatment for N=1 simpler than the earlier work. The Minkowski space in the current case, where the corresponding super Fuchsian groups act, is replaced by the superspace R2,2\4, and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in R2,2\4 as well as extra bosonic parameters, which we call ratios, defining a flat R+-connection on F. As in the pure bosonic or N=1 cases, we derive the analogue of Ptolemy transformations for all these new variables.

In this paper, we study the behavior of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.

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In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of RN with prescribed measure m attains its maximum on the union of two disjoint balls of measure m/2. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.