Differential Geometry and its Applications最新文献

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.

Here we obtain a classical integral formula on the conformal change of Finsler metrics. As an application, we obtain significant results depending on the sign of the Ricci scalars, for mean Landsberg surfaces and show there is no conformal transformation between two compact mean Landsberg surfaces, one of a non-positive Ricci scalar and another of a non-negative Ricci scalar, except for the case where both Ricci scalars are identically zero. Conformal transformations preserving the Ricci tensor are known as Liouville transformations. Here we show that a Liouville transformation between two compact mean Landsberg manifolds of isotropic S-curvature is homothetic. Moreover, every Liouville transformation between two compact Finsler n-manifolds of bounded mean value Cartan tensor is homothetic. These results are an extension of the results of M. Obata and S. T. Yau on Riemannian geometry and give a positive answer to a conjecture on Liouville's theorem.

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds.

The Chekanov torus is the first known exotic torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in $S^2\times S^2$ and a monotone Lagrangian torus that had been constructed before Chekanov's construction [6]. We prove that the monotone Lagrangian torus fiber in a certain Gelfand–Zeitlin system is related to the Chekanov torus in $S^2\times S^2$ by a symplectomorphism.

In this article, we investigate quasi-Einstein manifolds admitting a closed conformal vector field. Initially, we present a rigidity result for quasi-Einstein manifolds with constant scalar curvature and carrying a non-parallel closed conformal vector field. Moreover, we prove that quasi-Einstein manifolds admitting a closed conformal vector field can be conformally changed to constant scalar curvature almost everywhere. Finally, we obtain a characterization for quasi-Einstein manifolds endowed with a non-parallel gradient conformal vector field.

We consider complete locally irreducible conullity two Riemannian manifolds with constant scalar curvature along nullity geodesics. There exists a naturally defined open dense subset on which we describe the metric in terms of several functions which are uniquely determined up to isometry. In addition, we show that the fundamental group is either trivial or infinite cyclic.

An analogue of Brylinski's knot beta function is defined for a compactly supported (Schwartz) distribution T on d-dimensional Euclidean space. This is a holomorphic function on a right half-plane. If T is a (uniform) double-layer on a compact smooth hypersurface, then the beta function has an analytic continuation to the complex plane as a meromorphic function, and the residues are integrals of invariants of the second fundamental form. The first few residues are computed when $d=2$ and $d=3$.

I show that the S-curvature of a Finsler space vanishes if and only if the E-curvature vanishes if and only if the Berwald scalar curvature vanishes; and I extend these results to the case in which these objects are isotropic.