Differential Geometry and its Applications最新文献

We show that antipodal sets of pseudo-Riemannian symmetric R-spaces associated with non-degenerate Jordan triple systems satisfy the following two properties: (1) Any antipodal set is included in a great antipodal set, and (2) any two great antipodal sets are transformed into each other by an isometry.

In this paper, we study the prescribed fractional Q-curvatures problem of order 2σ on the n-dimensional standard sphere $(S^n,g_0)$, where $n\ge 3$, $\sigma \in (0,\frac{n-2}{2})$. By combining critical points at infinity approach with Morse theory we obtain new existence results under suitable pinching conditions.

Suppose $(M^4,g,f)$ is a complete shrinking gradient Ricci soliton. We give a sufficient condition for a soliton to be compact, generalizing previous result of Munteanu-Wang [17]. As an application, we give a classification of $(M^4,g,f)$ under some natural conditions.

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_{\omega }=d+\omega \wedge$, where ω is a closed 1-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology.

In this paper, we first prove that a quadratic form from $S^m$ to $S^n$ is non-harmonic biharmonic if and only if it has constant energy density $(m+1)/2$. Then, we give a positive answer to an open problem raised in [1] concerning the structure of non-harmonic biharmonic quadratic forms. As a direct application, using classification results for harmonic quadratic forms, we infer classification results for non-harmonic biharmonic quadratic forms.

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the lines principal curvature, the parabolic curve and the locus of points where the mean curvature vanishes.

In this paper, we prove a priori estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Ampère equation, prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles and get estimates for J-vortex equation. We prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove Kählerness of the negative of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations in [9].

Using the variational methods and the critical points theory, we prove the existence and the uniqueness of a positive solution for a singular Kirchhoff type equation on a closed Riemannian manifold of dimension $N\ge 3$. At the end, we give a geometric application involving the conformal Laplacian.

Let $(H,〈,〉)$ be a complex Hilbert space and $B\left(H\right)$ the space of bounded linear operators in $H$. Any other equivalent inner product in $H$ is of the form ${〈f,g〉}_A=〈Af,g〉$ ($f,g\in H$) for some positive invertible operator $A\in B\left(H\right)$. In this paper we study the bundle $M$ which consist of the unit sphere $\{f\in H:{〈f,f〉}_A=1\}$ over each (equivalent) inner product ${〈,〉}_A$, which due to the observation above can be defined$M=\left\{\right(A,f)\in B(H)\times H:A\text{ is positive and invertible and }〈Af,f〉=1\}.$ We prove that $M$ is a complemented submanifold of the Banach space $B\left(H\right)\times H$ and a homogeneous space of the Banach-Lie group $G\left(H\right)\subset B\left(H\right)$ of invertible operators. We introduce a reductive structure in $M$, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of $M$, for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C^⁎-algebra $A\subset B\left(H\right)$.

In this note, we consider the first nonzero eigenvalue ${\lambda }_{p,1}$ of the p-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that ${\lambda }_{p,1}$ is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of ${\lambda }_{p,1}$ of a free boundary disk type hypersurface in the unit ball.