This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.

Manifold learning is an effective dimensional reduction technique for face feature extraction, which, generally speaking, tends to preserve the local neighborhood structures of given samples. However, neighbors of a sample often comprise more inter-class data than intra-class data, which is an undesirable effect for classification. In this paper, we address this problem by proposing a subclass-...

Let M be a closed oriented 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straighten...

A b str act . Suppose N is an (n− 2)-manifold topologically embedded in an n-manifold M . Chapman and Quinn have shown that N is locally flat in M provided that the local homotopy groups of M − N at points of N are infinite cyclic and that all the higher local homotopy groups of M−N at points of N vanish. The main theorem in this paper asserts that it is only necessary to assume the homotopy co...

This note establishes smooth approximation from above for Jplurisubharmonic functions on an almost complex manifold (X, J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence uj ∈ C∞(X) of smooth strictly Jplurisub...

The classical result on local orbits in geometric manifolds is Singer’s homogeneity theorem for Riemannian manifolds [1]: given a Riemannian manifold M , there exists k, depending on dimM , such that if every x, y ∈ M are related by an infinitesimal isometry of order k, thenM is locally homogeneous. An open subset U ⊆ M of a geometric manifold is locally homogeneous if for every x, x ∈ U , ther...

We study generalized complex (GC) manifolds from the point of view of symplectic and Poisson geometry. We start by recalling that every GC manifold admits a canonical Poisson structure. We use this fact, together with Weinstein’s classical result on the local normal form of Poisson, to prove a local structure theorem for GC, complex manifolds, which extends the result Gualtieri has obtained in ...

The main result of this paper is that the space of conformally compact Einstein metrics on any given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric a...

LetM be an (n+1)-dimensional manifold with non-empty boundary, satisfying π1(M,∂M) = 0. The main result of this paper is that the space of conformally compact Einstein metrics on M is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with p...

A generic compact real codimension two submanifold X of Cn+2 will have a CR structure at all but a finite number of points (failing at the complex jump points J). The main theorem of this paper gives a method of extending the CR structure on the non-jump points X − J to the jump points. We examine a Gauss map from X − J to an appropriate flag manifold F and take the closure of the graph of this...