We study the CalabiYau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

1. Statement of results Let Mn be a compact manifold of dimension n endowed with a Riemannian metric. The spectrum of the Laplacian,, acting on functions form a discrete set of the form {0< 1 2 k}. In 1970, Joseph Hersch [5] gave a sharp upper bound for the first non-zero eigenvalue 1 for any Riemannian metric on the 2-sphere in terms of its volume alone. Similar estimates for 1 on any compact oriented surfaces were derived by Yang-Yau [7]. The second and the third authors [6] studied the nonorientable surfaces and pointed out the relationship of 1 and the conformal class of the surface. In fact, their estimates were applied to study the Willmore problem. Another application of these types of upper bounds was found by Choi-Schoen [3] in relation to the set of all minimal surfaces in a compact 3-manifold of positive Ricci curvature. The purpose of this paper is to prove a higher dimensional generalization of the above results. It was pointed out by Marcel Berger [1] that Herschs theorem fails in higher dimensional spheres. In view of the relationship between 1 and the conformal

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M-five-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into this framework. To give some examples, we construct two non-compact manifolds with G_2 structures: one with a K3 fibration, and one with a torus fibration and a metric of G_2 holonomy. Kaluza-Klein reduction of the latter solution gives abelian BPS monopoles in 3+ 1 dimensions.

In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ^<i>n</i> is an exotic<i>n</i>-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ^<i>n</i> are tori of dimension [(<i>n</i>+1)/2].