黎曼流形上混合边值问题的第一特征值估计

设M为一带边界M的紧致Riemann流形,本文考虑M上的下述混合边值条件的特征值问题 (△u+v_1u=0, u/n+αu|M=0,)其中n为M的外法向单位向量,α为一正常数。     

Residue Traces for a Pseudodifferential Operator Algebra on Foliated Manifolds

In this paper a trace functional is defined for the algebra of classical anisotropic pseudo-differential operators on a compact foliated manifold. Furthermore, by means of the Weyl formula, a formula is proved specifying the link between this trace and Dixmier's trace. Submitted: June 11, 2001?Revised: February 17, 2002     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

Resolvent Estimates for the Laplacian on Asymptotically Hyperbolic Manifolds

Combining results of Cardoso-Vodev [6] and Froese-Hislop [9], we use Mourre’s theory to prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. We derive estimates involving a class of pseudo-differential weights which are more natural in the asymptotically hyperbolic geometry than the weights used in [6]. submitted 28/04/05, accepted 26/09/05     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

Eigenvalue estimate for the rough Laplacian on differential forms

In this article, we study the spectrum of the rough Laplacian acting on differential forms on a compact Riemannian manifold (M, g). We first construct on M metrics of volume 1 whose spectrum is as large as desired. Then, provided that the Ricci curvature of g is bounded below, we relate the spectrum of the rough Laplacian on 1-forms to the spectrum of the Laplacian on functions, and derive some upper bound in agreement with the asymptotic Weyl law.     

Zeta Determinants on Manifolds with Boundary

Through a general theory for relative spectral invariants, we study the ζ-determinant of global boundary problems of APS-type. In particular, we compute the ζ-determinant ratio for Dirac-Laplacian boundary problems in terms of a scattering Fredholm determinant over the boundary.     

Ricci-Bourguignon Flow on Manifolds with Boundary

The authors consider the short time existence for Ricci-Bourguignon flow on manifolds with boundary. If the initial metric has constant mean curvature and satisfies some compatibility conditions, they show the short time existence of the Ricci-Bourguignon flow with constant mean curvature on the boundary.     

Quantization Formula for Symplectic Manifolds with Boundary

We extend our earlier work in [TiZ1], where an analytic approach to the Guillemin-Sternberg geometric quantization conjecture [GuSt] was developed, to the case of manifolds with boundary. We also give a general quantization formula that works for both regular and singular reductions. As simple applications, we prove an analytic analogue of the relative residue formula of Guillemin-Kalkman [GuK] and Martin [M], as well as a Guillemin-Sternberg type formula for singular reductions under circle actions. Submitted: February 1997, revised: January 1998 and July 1998, final version: March 1999.     

带权流形上热方程的Harnack估计

利用两种不同的方法讨论了带权流形上热方程和Schrodinger方程解的Harnack估计,先利用最大模原理证明热方程解的梯度估计,从而得到解的Harnack估计,另外利用算子半群的方法证明位势函数为常数的Schrodinger方程解的Harnack估计.     

Eigenvalue bounds of the Robin Laplacian with magnetic field

On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues \((\lambda _k(\tau ,\alpha ))_k\) of the magnetic Laplacian with Robin boundary conditions. Here, \(\tau \) is a positive number that defines the Robin condition and \(\alpha \) is a real differential 1-form on M that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter \(\tau \), and a lower bound of the Ricci curvature of M (see Theorem 1.3 and Corollary 1.5). The main technique is to use the Bochner formula established in Egidi et al. (Ricci curvature and eigenvalue estimates for the magentic Laplacian on manifolds, arXiv:1608.01955v1) for the magnetic Laplacian and to integrate it over M (see Theorem 1.2). In the last part, we compare the eigenvalues \(\lambda _k(\tau ,\alpha )\) with the first eigenvalue \(\lambda _1(\tau )=\lambda _1(\tau ,0)\) (i.e. without magnetic field) and the Neumann eigenvalues \(\lambda _k(0,\alpha )\) (see Theorem 1.6) using the min-max principle.     

Inverse Problems for Differential Forms on Riemannian Manifolds with Boundary

Consider a real-analytic orientable connected complete Riemannian manifold M with boundary of dimension n ≥ 2 and let k be an integer 1 ≤ k ≤ n. In the case when M is compact of dimension n ≥ 3, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic k-forms, given on an open subset of the boundary. This extends a result of [14 Lassas , M. , Uhlmann , G. ( 2001 ). On determining a Riemannian manifold from the Dirichlet-to-Neumann map . Ann. Sci. École Norm. 34 : 771 – 787 .[Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic 1-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when M is complete non-compact. In the case n ≥ 3, this generalizes the results of [13 Lassas , M. , Taylor , M. , Uhlmann , G. ( 2003 ). The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary . Comm. Anal. Geom. 11 : 207 – 221 .[Crossref], [Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic 1-forms.     

带权Laplacian方程解的最优梯度估计

设M为n维完备无边界的流形,它的Ricci曲率有下界-K,这里K为实常数.假设M上的向量场B满足|B|≤γ且(△)B≤K*,这里γ为非负常数,K*为实常数,则带权Laplacian方程△u Bu=0任意正的光滑解满足最优梯度估计|(△u)|2/u2≤m(K K*) mγ2/m-n,其中任意常数m＞n.     

Mass-Capacity Inequalities for Conformally Flat Manifolds with Boundary

In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, capacity and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is characterized.     

Eigenvalue inequalities for the buckling problem of the drifting Laplacian

In this paper, we study the buckling problem of the drifting Laplacian on bounded domains in a complete Riemannian manifold with nonnegative ∞-dimensional Bakry–Émery Ricci curvature. According to the property of the manifold, we obtain a family of trial functions. By making use of these trial functions, we derive a universal inequality of eigenvalues, which is independent of the domains.     

Modified Curvatures on Manifolds with Boundary and Applications

To study the reflecting diffusion processes on manifolds with boundary, some new curvature operators are introduced by using the Bakry-Emery curvature and the second fundamental form. As applications, the gradient estimates, log-Harnack inequality and Poincaré/log-Sobolev inequalities are investigated for the Neumann semigroup on manifolds with boundary.     

Bounds for the Least Laplacian Eigenvalue of a Signed Graph

A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.