We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.

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Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0 > nKmax, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if Mn(n  3) is a compact manifold whose (n . 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric(n.2) min > n(n .2)R0, where n 2 ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker, and the authors.

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rigid Calabi-Yau manifolds poses a serious challenge to the conjecture. The proposed mirror partners for them are higher dimensional generalized Calabi-Yau manifolds. For example, the mirror partner for a certain K3 surface is a cubic fourfold and its Fano variety of lines is birational to the Hilbert scheme of two points on the K3. This hyperk¨ahler manifold can be interpreted as the SYZ mirror of the K3 by considering singular special Lagrangian tori. We also compare the geometries between a CY and its associated generalized CY. In particular we present a new construction of Lagrangian submanifolds.

Evolving smooth, compact hypersurfaces in Rn+1 with normal speed equal to a positive power k of the mean curvature improves a certain ‘isoperimetric difference’ for k > n.1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n 6 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.