p-harmonic 1-forms on complete manifolds

Let (M ^m , g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L ^q (M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

General estimate of the first eigenvalue on manifolds

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.     

Closed holomorphic 1-forms without zeros on Stein manifolds

We prove that a stein mainfold admits a closed holomorphic 1-form without zeros in every class of the first cohomology group. We also prove an approximation result for closed holomorphic 1-forms without zeros defined in a neighborhood of a compact subset.     

A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.     

Linear approximation of the first eigenvalue on compact manifolds

For compact, connected Riemannian manifolds with Ricci curvature bounded below by a constant, what is the linear approximation of the first eigenvalue of Laplacian? The answer is presented with computer assisted proof and the result is optimal in certain sense.     

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.     

Gradient estimates and the first Neumann eigenvalue on manifolds with boundary

By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on non-convex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via dimension, radius and bounds of the curvature and the second fundamental form. Finally, some new estimates are also presented for the strictly convex case.     

The first nonzero eigenvalue of Neumann problem on Riemannian manifolds

In this article, the author obtained some comparison theorems of the first nonzero Neumann eigenvalue on domains in nonpositively curved Riemannian manifolds. The author first gives a generalized Szegö-Weinberger theorem (Theorem 1). Then the first nonzero Neumann eigenvalues for geodesic balls on nonpositively curved Riemannian manifolds are compared (Theorem 2). Based on these results, a “monotonicity principle” for the Neumann eigenvalues is derived. Then the author proves a stability theorem of maximality of the first nonzero Neumann eigenvalue of a geodesic ball among those of all domains with the same volume.     

Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

Given a manifold \(M\) , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.     

The first eigenfunctions and eigenvalue of the p-Laplacian on Finsler manifolds

This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C~(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.     

On the first eigenvalue of Finsler manifolds with nonnegative weighted Ricci curvature

We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.     

The natural operators lifting 1-forms on manifolds to the bundles ofA-velocities

LetA be a Weil algebra withp variables. We prove that forn-manifolds (np+2) the set of all natural operatorsT ^*T ^* T ^A is a free finitely generated module over a ring canonically dependent onA. We construct explicitly the basis of this module.     

On 3-dimensional asymptotically harmonic manifolds

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature , provided M is asymptotically harmonic of constant h > 0. Received: 4 October 2007     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.