Теоремы о дифференцируемых сферах для многообразий с ограниченными сверху кривизнами Риччи

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic points needed to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem. 1. Dynamical and geometrical assumptions Let M be a compact Riemannian manifold of dimension d ≥ 1 with a normalized Riemannian volume | · |, which we call Lebesgue measure. Let f : M → M be a C local diffeomorphism for all x ∈ M C, where C is some critical set, which may include points at which the derivative Dfx is degenerate, as well as points of discontinuity and points at which the derivative is infinite. We assume the following natural non-degeneracy condition on C, which generalizes the notion of non-flat critical points for smooth one-dimensional maps. Definition 1. The critical set C ⊂ M is non-degenerate if |C| = 0 and there is a constant β > 0 such that for every x ∈M C we have dist(x, C) . ‖Dfxv‖/‖v‖ . dist(x, C)−β for all v ∈ TxM , and the functions log detDf and log ‖Df−1‖ are locally Lipschitz with Lipschitz constant . dist(x, C)−β . We now state our two dynamical assumptions: the first is on the growth of the derivative and the second is on the approach rate of orbit to the critical set. Notice that for a linear map A, the condition ‖A‖ > 1 only provides information about the existence of some expanded direction, whereas the condition ‖A−1‖ 0) implies that every direction is expanded. Received by the editors November 5, 2002. 2000 Mathematics Subject Classification. Primary 37D20, 37D50, 37C40. Work carried out at the Federal University of Bahia, University of Porto and Imperial College, London. Partially supported by CMUP, PRODYN, SAPIENS and UFBA. c ©2003 American Mathematical Society

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in $$n$$ -dimensional compact Riemannian manifolds for $$n=2,3$$ . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- $$\alpha $$ -like model, the Leray- $$\alpha $$ model, the modified Leray- $$\alpha $$ model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity $$u$$ suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter $$\phi $$ . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the $$\alpha \rightarrow 0$$ limit in $$\alpha $$ models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory $$\left( u,\phi \right) $$ converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in $$n$$ -dimensional compact Riemannian manifolds.

An isometric immersion of an $n$-dimensional compact Riemannian manifold with sectional curvature always less than ${\lambda ^{ - 2}}$ into Euclidean space of dimension $2n - 1$ can never be contained in a ball of radius $\lambda$. This generalizes and includes results of Tompkins and Chern and Kuiper.

Let (M,g) be a four dimensional compact Riemannian manifold with boundary and (N,h) be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from M to N under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from M to N under any Dirichlet boundary condition.

Exponential problem on a compact Riemannian manifold without boundary

The vorticity is important in most of the fluid problems. Also, two dimensional manifolds are especially important in fluid mechanics. In this paper, we use the geometric analysis to obtain vorticity as the flow on the any 2–dimensional compact Riemannian manifold (M, g). Especially, we explain vorticity flow on the 2–dimensional sphere.

Sharp bounds for the first non-zero Stekloff eigenvalues

Sharp Nash inequalities on manifolds with boundary in the presence of symmetries

Let M be an n -dimensional compact Riemannian manifold with a boundary. In this paper, we consider the Steklovrst eigenvalue with respect to the f -divergence form:

We study the existence or non-existence of harmonic diffeomorphisms between certain domains in the Euclidean 2 -sphere. In particular, we show the existence of harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product \mathbb{M}\times\mathbb{R}_1 , where \mathbb{M} is an arbitrary \mathfrak{n} -dimensional compact Riemannian manifold, \mathfrak{n}\geq 2 . In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once-punctured sphere and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean 2 -sphere.

Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach

Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

<p style='text-indent:20px;'>We establish Strichartz estimates for the regularized Schrödinger equation on a two dimensional compact Riemannian manifold without boundary. As a consequence we deduce global existence and uniqueness results for the Cauchy problem for the nonlinear regularized Schrödinger equation and we prove under the geometric control condition the Kato smoothing effect for solutions of this equation in this particular geometries.

Journal Article A Remark on Zhong-Yang's Eigenvalue Estimate Get access Fengbo Hang, Fengbo Hang 1 Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544 Search for other works by this author on: Oxford Academic Google Scholar Xiaodong Wang Xiaodong Wang 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824 Correspondence to be sent to: Xiaodong Wang, Department of Mathematics, Michigan State University, East Lansing, MI 48824, e-mail: xwang@math.msu.edu Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2007, 2007, rnm064, https://doi.org/10.1093/imrn/rnm064 Published: 01 January 2007 Article history Received: 06 November 2006 Published: 01 January 2007 Revision received: 03 March 2007 Accepted: 07 July 2007

Author(s): Paternain, GP; Salo, M; Uhlmann, G; Zhou, H | Abstract: Consider a compact Riemannian manifold of dimension ≥ 3 with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension ≥ 3 having nonnegative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.