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Qiao-hua Yang - One of the best experts on this subject based on the ideXlab platform.
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An interpolation of Hardy inequality and Moser–Trudinger inequality on Riemannian Manifolds with negative curvature
Acta Mathematica Sinica, 2016Co-Authors: Yan Qing Dong, Qiao-hua YangAbstract:Let M be a complete, simply Connected Riemannian Manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser–Trudinger inequality on M. Furthermore, the constant we obtain is sharp.
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Sharp Moser–Trudinger inequalities on Riemannian Manifolds with negative curvature
Annali di Matematica Pura ed Applicata, 2015Co-Authors: Qiao-hua Yang, Dan Su, Yinying KongAbstract:Let \(M\) be a complete, simply Connected Riemannian Manifold with negative curvature. We obtain some Moser–Trudinger inequalities with sharp constants on \(M\).
Martin N. Ndumu - One of the best experts on this subject based on the ideXlab platform.
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An integral formula for the heat kernel of tubular neighborhoods of complete (Connected) Riemannian Manifolds
Potential Analysis, 1996Co-Authors: Martin N. NdumuAbstract:Let M be a complete Connected Riemannian Manifold and let N be a subManifold of M. Let v: Ev»N be the normal bundle of N and expv: Ev»M its exponential map.
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An integral formula for the heat kernel of tubular neighborhoods of complete (Connected) Riemannian Manifolds
Potential Analysis, 1996Co-Authors: Martin N. NdumuAbstract:Let M be a complete Connected Riemannian Manifold and let N be a subManifold of M. Let v : E _v»N be the normal bundle of N and exp_ v : E _v»M its exponential map. Let (exp _inf v ^/sup-1 , M _0) be the Fermi chart relative to the subManifold N . Then, by using the Fermi coordinates we obtain an integral formula for the Dirichlet heat kernel p _t ^m (-,-). That is, we obtain a probabilistic representation for the integral ∫_N f ( y )p _t ^M (x,y) d y where f is any measurable function of compact support in M _0. This representation involves a subManifold semi-classical Brownian Riemannian bridge process. Then applying the integral formula via a Riemannian submersion in [5], we obtain heat kernel formulae for the complex projective space cP ^n, the quaternionic projective space Q P ^n and the Caley line Ca P ^1. The case of the Caley plane Ca P ^2 eludes us due to the lack of a submersion theorem.
Yinying Kong - One of the best experts on this subject based on the ideXlab platform.
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Sharp Moser–Trudinger inequalities on Riemannian Manifolds with negative curvature
Annali di Matematica Pura ed Applicata, 2015Co-Authors: Qiao-hua Yang, Dan Su, Yinying KongAbstract:Let \(M\) be a complete, simply Connected Riemannian Manifold with negative curvature. We obtain some Moser–Trudinger inequalities with sharp constants on \(M\).
Hironori Kumura - One of the best experts on this subject based on the ideXlab platform.
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Spectral Convergence of Conformally Immersed Surfaces with Bounded Mean Curvature
Journal of Geometric Analysis, 2002Co-Authors: Atsushi Kasue, Hironori KumuraAbstract:Motivated by B6rard, Besson, and Gallot [1], we introduced in [10] and [11] a distance between compact Connected Riemannian Manifolds using their heat kernels and studied its basic properties. The distance is called the spectral distance SD and defined as follows: For a compact Connected Riemannian Manifold (M, g), we denote by PM (t, x, y) the heat kernel of the Laplace operator of M with respect to the normalized Riemannian measure IZM = dvg/Vol(M, g). Given two compact Connected Riemannian Manifolds M and N, a map f : M --~ N is called an s-spectral approximating map if it satisfies
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SPECTRAL CONVERGENCE OF Riemannian ManifoldS, II
Tohoku Mathematical Journal, 1994Co-Authors: Atsushi Kasue, Hironori KumuraAbstract:We prove a precompactness theorem concerning the spectral distance on the set of isometry classes of compact Riemannian Manifolds and study the comple- tion of a precompact family. Introduction. For a compact Connected Riemannian Manifold M=(M,g), we denote by pM(t, x, y) the heat kernel of the Laplace operator of M with respect ot the normarized Riemannian measure μM (= dvg/Vol(M)). Given two compact Connected Riemannian Manifolds M and N, a mapping /: M->N is called an e-spectral ap- proximation if it satisfies e-{t + llt) \pM(U x, y)~PN(t, f(x)9 Ry)) I 0 and x,yeM. The spectral distance SD(M, N) between M and N is by definition the lower bound of the positive numbers e such that there exist e-spectral approximations /: M-^N and h: N^M. The distance SD gives a uniform structure on the set Mc of isometry classes of compact Connected Riemannian Manifolds. Riemannian Manifolds are considered as metric spaces endowed with Riemannian distances. From this point of view, the set Mc has another uniform structure introduced by Gromov (18), called the Hausdorff distance HD. In (18), the conditions for a family of Jic to be HD-precompact are described and it is shown that the boundaries of such a family consist of certain metric spaces, called length spaces. This decade has seen intensive activities around the convergence theory of Riemannian Manifolds with respect to the Gromov-Hausdorff distance. These includes some works from the viewpoint of spectral geometry, for instance, (14), (4), and (23). In (25), motivated by these results, we introduced the spectral distance SD mentioned above and discussed some basic properties of the distance on a set of compact Connected Riemannian Manifolds of the same dimension with diameters uniformly bounded from above and Ricci curvatures uniformly bounded from below. In the present paper, we are concerned with a certain precompact family of Jίc and its compactification with respect to the spectral distance. More precisely, the main results are stated as follows.
Matti Lassas - One of the best experts on this subject based on the ideXlab platform.
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Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem
New Analytic and Geometric Methods in Inverse Problems, 2020Co-Authors: Atsushi Katsuda, Yaroslav Kurylev, Matti LassasAbstract:In this paper we study stability and approximate reconstruction in the inverse boundary spectral problem (the generalized Gelfand inverse problem [12]) for Riemannian Manifolds. We denote by (M, g) an unknown, m-dimensional, compact Connected Riemannian Manifold with a (smooth) metric g and non-empty boundary ∂M. The boundary ∂M is itself an (m − 1)-dimensional compact differentiable Manifold. We do not assume the knowledge of i* (g) on ∂M, where i : ∂M → M is an embedding or the corresponding area element dS g . Because the boundary ∂M is known, we will consider a class M = M ∂M of compact, Connected Riemannian Manifolds which have the same boundary, ∂M.
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Inverse spectral problems on a closed Manifold
J MATH PURE APPL, 2008Co-Authors: Matti LassasAbstract:In this paper we consider two inverse problems on a closed Connected Riemannian Manifold (M, g). To formulate the first one, assume that M is divided by a hypersurface Sigma into two components and we know the eigenvalues lambda(j) of the Laplace operator on (M, g) and also the Cauchy data, on Sigma, of the corresponding eigenfunctions phi(j), i.e. phi j vertical bar(Sigma), partial derivative(nu)phi(j)vertical bar(Sigma), where nu is the normal to Sigma. We prove that these data determine (M, g) uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, lambda(j) and phi(j)vertical bar(Sigma) only. However, if Sigma consists of at least two components, Sigma(1), Sigma(2), we are still able to determine (M, g) assuming some generic conditions on the spectra of the Laplacian in subdomains of M obtained by cutting along Sigma. (C) 2008 Elsevier Masson SAS. All rights reserved.