We study minimizers of the functionalwhere $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$ ,$u$=0 on {$x$∈$B$_{1}:$x$_{1}=0}, $\lambda^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .1ex\hbox {$\scriptscriptstyle \pm $}} {\scriptscriptstyle \pm }}}$ are two positive constants and 0<$p$<1. In two dimensions, we prove that the free boundary is a uniform$C$^{1}graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {$x$:±$u$($x$)>0}.

It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface f admits singularities but its Gauss map  is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.

This is the very first paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.

The main purpose of this paper is to prove a general result about the geometry of holomorphic line bundles over Khler manifolds. This result is essentially a partial verification of a conjecture of Tian [30] and Tian has, over many years, highlighted the importance of the question for the existence theory of KhlerEinstein metrics. We will begin by stating this main result.

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in our previous paper [HL4], we define equations via closed subsets of the 2-jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Amp`ere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain  X, we give geometric conditions which imply that the Dirichlet problem on is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then introduce two fundamental concepts. The first is the notion of a monotonicity cone M for F. If X carries a global Msubharmonic function, then weak comparison implies full comparison. The second notion is that of boundary F-convexity, which is defined in terms of the asymptotics of F and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed. This article also introduces the notion of local affine jet-equivalence for subequations. It is used in treating the cases above, but gives results for a much broader spectrum of equations on manifolds, including inhomogeneous equations and the Calabi-Yau equation on almost complex hermitian manifolds. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.