We introduce a surgery for generalized complex manifolds whose input is a symplectic 4-manifold containing a symplectic 2-torus with trivial normal bundle and whose output is a 4-manifold endowed with a generalized complex structure exhibiting type change along a 2-torus. Performing this surgery on a K3 surface, we obtain a generalized complex structure on 3CP2#19CP2, which has vanishing Seiberg–Witten invariants and hence does not admit complex or symplectic structures.

We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston’s gluing equation and Haken’s normal surface equation. The action functional is the volume.

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.

No abstract uploaded!

For Riemannian manifolds with a measure (M, g, e.fdvolg) we prove mean curvature and volume comparison results when the亣- Bakry-Emery Ricci tensor is bounded from below and f or |f| is bounded, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.