Let ($M$,$ω$) be a connected, symplectic 4-manifold. A$semitoric integrable system$on ($M$,$ω$) essentially consists of a pair of independent, real-valued, smooth functions$J$and$H$on$M$, for which$J$generates a Hamiltonian circle action under which$H$is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors, this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants.

In this article, we study the Hamiltonian non-displaceability of Gauss images of isopara- metric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem which applies only to calibrated submanifolds of special holonomy ambient manifolds.

We define analytic torsion (X, E,H) 2 detH•(X, E,H) for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle E, with a differential given by rE + H ^ · , where rE is a flat connection on E, H is an odd-degree closed differential form on X, and H•(X, E,H) denotes the cohomology of this Z2-graded complex. The definition uses pseudodifferential operators and residue traces. We show that when dimX is odd, (X, E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H 7! H−dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We demonstrate some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart, and prove an analogue of the Cheeger-M¨uller Theorem. We also study the twisted analytic torsion for T -dual circle bundles with integral 3- form fluxes.

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