Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperk¨ahler manifold, up to finite group factor.

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau’s inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.

The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-K¨ahler manifolds with torsion