Let $f$ be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of $f$ under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is isotopic to a constant map.

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There have been many attempts to define the notion of quasilocal mass for a spacelike 2-surface in spacetime by the Hamilton-Jacobi analysis. The essential difficulty in this approach is to identify the right choice of the background configuration to be subtracted from the physical Hamiltonian. Quasilocal mass should be nonnegative for surfaces in general spacetime and zero for surfaces in flat spacetime. In this letter, we propose a new definition of gauge-independent quasilocal mass and prove that it has the desired properties.

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.

The dimension datum of a subgroup of a compact Lie group is a piece of spectral information about that subgroup. We find some new invariants and phenomena of the dimension data and apply them to construct the first example of a pair of isospectral, simply connected closed Riemannian manifolds which are of different homotopy types. We also answer questions proposed by Langlands.