We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

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 give existence results for simple closed curves with prescribed geodesic curvature on S2, which correspond to periodic orbits of a charge in a magnetic field.

We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of π1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats