I. Introduction. Stokes' theorem on a manifold is a central theorem of mathematics. Special cases include the integral theorems of vector analysis and the Cauchy-Goursat theorem. My purpose here is to prove this version of Stokes' Theorem. Let ω be a continuous differential (n − 1)-form on a compact oriented n-manifold M with boundary ∂M. Suppose that ω is differen-tiable on M − ∂M and dω is Le...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

We will simplify the earlier proofs of Perelman’s collapsing theorem of 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states that: “Let {M3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M3 i ) ≥ c0 > 0. Suppose that all unit metric balls in M3 i have very small volum...

Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this manifold. This reduced system may provide valuable dynamical information for the original system. The authors have derived an invariant manifold reduction or r...

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya-Yamaguchi [SY00]-[SY05] and Morgan-Tian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M3 i } be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and diam(M3 i ) ≥ c0 > 0. Suppose that all unit metric balls in M3 i have very small volume at mos...

Let M be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on M , a local Fatou Theorem and a pointwise criteria of nontangential convergence coming from the density of energy are shown: at almost all points of the boundary, the harmonic function converges non-tangentially if and only if the supremum of the ...

In [No], Nori studied the fundamental group of complements of nodal curves with ample normal bundle in smooth projective surfaces. The main tool was the following weak Lefschetz theorem: Theorem (Nori). Suppose Φ : U → X is a local biholomorphism from a connected complex manifold U into a connected smooth projective variety X of dimension at least 2, and U contains a connected effective divisor...

In [No], Nori studied the fundamental group of complements of nodal curves with ample normal bundle in smooth projective surfaces. The main tool was the following weak Lefschetz theorem: Theorem (Nori). Suppose Φ : U → X is a local biholomorphism from a connected complex manifold U into a connected smooth projective variety X of dimension at least 2, and U contains a connected effective divisor...