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1.
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.  相似文献   

2.
We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].  相似文献   

3.
We construct an infinite family of hyperbolic three-manifolds with geodesic boundary that generalize the Thurston and Paoluzzi-Zimmermann manifolds. For the manifolds of this family, we present two-sided bounds for their complexity.  相似文献   

4.
We construct Gauss–Weingarten-like formulas and define O’Neill’s tensors for Riemannian maps between Riemannian manifolds. By using these new formulas, we obtain necessary and sufficient conditions for Riemannian maps to be totally geodesic. Then we introduce semi-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, give examples and investigate the geometry of leaves of the distributions defined by such maps. We also obtain necessary and sufficient conditions for semi-invariant maps to be totally geodesic and find decomposition theorems for the total manifold. Finally, we give a classification result for semi-invariant Riemannian maps with totally umbilical fibers.  相似文献   

5.
In this work we extend the idea of warped products, which was previously defined on smooth Riemannian manifolds, to geodesic metric spaces and prove the analogue of the theorems on spaces with curvature bounded from above.

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6.
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan–Hadamard spaces of strictly negative curvature, by the existence of an asymptotic equivalent \(\tau (u)\mathrm {e}^{Er}\) for the volume-density of geodesic spheres (with \(\tau \) constant in case \(DR_M\) is bounded). Finally, we show the existence of a Margulis function, and explicitly compute it, for all asymptotically harmonic manifolds.  相似文献   

7.

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern–Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov–Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

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8.
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.  相似文献   

9.
We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.  相似文献   

10.
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson–Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.  相似文献   

11.
In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.  相似文献   

12.
In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry, and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.  相似文献   

13.
We establish the equivalence between the family of uniformly regular Riemannian manifolds without boundary and the class of manifolds with bounded geometry.  相似文献   

14.
This paper studies manifolds-with-boundary collapsing in the Gromov– Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The first group of results provide a fiber bundle structure to the manifolds-with-boundary. One of the main theorems establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov–Hausdorff topology to a closed manifold. Another result is a rough version of Toponogov’s Splitting Theorem. The second group of results identify Gromov–Hausdorff limits of certain sequences of manifolds with non-convex boundaries as Alexandrov spaces of curvature bounded below.  相似文献   

15.
Große  Nadine  Nistor  Victor 《Potential Analysis》2020,53(2):407-447
Potential Analysis - We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of...  相似文献   

16.
The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.  相似文献   

17.
In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.  相似文献   

18.
一类完备Riemann流形上的有界调和函数   总被引:2,自引:0,他引:2  
王晓辉 《数学学报》1995,38(2):171-181
本文我们将对一类完备Riemann流形上的有界调和函数所组成的线性空间的维数的上界进行估计,同时给出了一个关于测地球体积的Bishop-Gromov型体积比较定理。  相似文献   

19.
The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.  相似文献   

20.
We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.  相似文献   

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