Remark about the spectrum of the p-form Laplacian under a collapse with curvature bounded below

Collapsing and the Differential Form Laplacian : The Case of a Smooth Limit Space

We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence.

Collapsing and the Differential Form Laplacian : The Case of a Singular Limit Space

In this paper, which is a sequel to math.DG/9902111, we analyze the limit of the p-form Laplacian under a collapse with bounded sectional curvature and bounded diameter to a singular limit space. As applications, we give results about upper and lower bounds on the j-th eigenvalue of the p-form Laplacian, in terms of sectional curvature and diameter.

关于双曲空间中极小超曲面谱的注记

Remarks on the spectra of minimal hypersurfaces in the hyperbolic space

本文计算了双曲空间中具有给定渐近数据的奇异面积最小化超曲面的Laplacian谱，推广至高余维数，并探讨了谱底部的极值性质。

Eigenvalues of collapsing domains and drift Laplacians

By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to gen…

关于具有全测地纤维的拉普拉斯算子与黎曼淹没的注记

Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers

本文研究具有全测地纤维的黎曼淹没的典型变分中拉普拉斯算子的第一正特征值，在纤维为Einstein且Ricci曲率满足条件时得到特征值界，并应用于Yamabe泛函临界点稳定性及四元数Kähler流形的twistor纤维化。

Eigenvalues of Xin-Laplacian on Complete Riemannian manifolds

In this paper, we firstly consider Dirichlet eigenvalue problem which is related to Xin-Laplacian on the bounded domain of complete Riemannian manifolds. By establishing the general formulas, combining with some results of Chen and Cheng type, we prove some eigenvalue inequalities. As some applications, we consider the eigenvalues on some Riemannian manifolds admitting with special functions, the translating solitons, minimal submanifolds on the Euclidean spaces, submanifolds on the unit spheres, projective spaces and so on. In particular, for the case of translating solitons, some eigenvalue …

Some remarks on sub-elliptic equations

In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on $R^n$. We also study a related semi-linear sub-elliptic equation in the whole $R^n$ and prove that under a suitable condition, we have infinite many positive solutions of the problem.

Remark about scalar curvature and Riemannian submersions

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning the scalar curvature of a smooth limit space arising in a bounded curvature collapse.

Remarks on non-compact gradient Ricci solitons

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and $L^p$-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under $L^p$ conditions on the relevant quantities.

Low eigenvalues and one-dimensional collapse

Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show tha…

Remarks on curvature in the transportation metric

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation $e^Φ = \det D^2 Φ$ on proper convex cones. We prove a generalization of this theorem by showing that for every $Φ$ solving this equation on a proper convex domain $Ω$ the corresponding metric measure space $(D^2 Φ, e^Φdx)$ has a non-positive Bakry-{É}mery tensor. Modifying the Calabi computations we obtain this result by ap…

Petite valeurs propres des fibrés principaux en tores

Let M^n be a compact n-dimensional principal T^k-bundle. We consider collapsings of M on N=M/T^k such that the diameter and sectional curvature of M satisfy diam(M)<d and |K(M)|<a, and give examples of collapsings for all k such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of M are bounded above by c(M).inj(M)^2k. Moreover, we prove that the first non-zero eigenvalue of 1-form Laplacian of all T^k-bundle M over N is bounded below by c(n,d,a,N).Vol(M)^2 and c.inj(M)^2k when M collapses on N.

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces): - we develop an intrinsic theory of Laplacian bounds in viscosity sense and in a pointwise, heat flow related, sense, showing their equivalence also with Laplacian bounds in distributional sense; - relying on these tools, we establish a PDE principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the $p$-Hopf-Lax semigroup, for general exponents $p\in[1,\infty)$. This principle adm…

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Lap…

Resolvents and Martin boundaries of product spaces

In this paper we examine the Laplacian on the product of two asymptotically hyperbolic (or conformally compact, as they are often called) spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolvent kernel itself. We use these results to find the Martin boundary of the product space. This behaves (nearly) as expected when the factors have no $L^2$ eigenvalues, but it experiences a substantial collapse in the presence of such eigenvalues.