Estimates for lower order eigenvalues of a clamped plate problem

For a bounded domain Ω in a complete Riemannian manifold M ⁿ , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.     

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L ²-analysis.Dedicated to Professor Hajime Urakawa on his sixtieth birthday.The author is partially supported by the Grant-in-Aid for Scientific Research No. 16740026 of the Japan Society for the Promotion of Science.     

Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013). Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili (Int Math Res Not 17:939–952, 1999) which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.     

Generalized Einstein tensor for a Weyl manifold and its applications

It is well known that the Einstein tensor G for a Riemannian manifold defined by G βα = R βα 1/2 Rδβα , R βα = g βγ R γα where R γα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

Volume comparison and the σk-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.     

Sub-Laplacian Eigenvalue Bounds on CR Manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjöstrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.     

Isoperimetric control of the Steklov spectrum

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

Conformally flat Lorentzian hypersurfaces in ℝ<Stack><Subscript>1</Subscript><Superscript>4</Superscript></Stack> with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x: M ₁ ³ → ? ₁ ⁴ is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of ? ₁ ⁴ . Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.     

The second Yamabe invariant

Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.     

Eigenvalue estimates for Dirac operators with parallel characteristic torsion

Assume that the compact Riemannian spin manifold (Mn,g) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion (∇T=0), and consider the Dirac operator D1/3 corresponding to the torsion T/3. This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D1/3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.     

Curvature-free upper bounds for the smallest area of a minimal surface

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mⁿ with a trivial first homology group. The first upper bound will be in terms of the diameter of Mⁿ, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mⁿ. If n = 3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mⁿ with . The above results are the first results of such nature. Received: October 2004 Revision: May 2005 Accepted: June 2005     

Estimates for lower order eigenvalues of quadratic polynomials of the Laplacian

Extending the results of Cheng et al. [8], we study eigenvalues of lower order of quadratic polynomial of the Laplacian on a bounded domain in a complete Riemannian manifold and obtain sharp universal inequalities for them.     

Riemannian Foliations and Eigenvalue Comparison

Eigenvalue comparison theorems for the Laplacian on a Riemannian manifold generally give bounds for the first Dirichlet eigenvalue on balls in the manifold in terms of an eigenvalue arising from a geometrically or analytically simpler situation. Cheng's eigenvalue comparison theory assumes bounds on the curvature of the manifold and then compares this eigenvalue to the eigenvalue of a ball in a constant curvature space form. In this paper we examine the basic Laplacian – the appropriate Laplacian on functions that are constant on the leaves of the foliation. The main theorems generalize Cheng's eigenvalue comparison theorem and other eigenvalue comparison theorems to the category of Riemannian foliations by estimating the first Dirichlet eigenvalue for the basic Laplacian on a metric tubular neighborhood of a leaf closure. Several other facts about the the first eigenvalue of such foliated tubes as well as some needed facts about the tubes themselves are established. This comparison theory, like Cheng's theorem, remains valid for large tubes that are not homotopic to the middle leaf closure and that may have irregular boundaries. We apply these results to obtain upper bounds for the eigenvalues of the basic Laplacian on a closed manifold in terms of curvature bounds and the transverse diameter of the foliation.     

The Conformal Willmore Functional: A Perturbative Approach

The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds (M,g)) is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds (?³,g _? )—where g _? is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds (M,g) of dimension three.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz算子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界估计.     

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. The Osserman Conjecture asserts that any Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in ${\mathbb {R}^{16}}$ . As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space is conformally equivalent to it.