scholarly journals (Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n

2019 ◽  
Vol 09 (01) ◽  
pp. 1950003
Author(s):  
Tianyi Ren
Keyword(s):  
Constant Curvature ◽  
Spherical Harmonics ◽  
Resolvent Estimate ◽  
Uniform Bound ◽  
Resolvent Estimates ◽  
Essential Ingredient ◽  
Norm Estimate ◽  
Interpolation Technique ◽  
Simply Connected ◽  
Funct Anal

We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

DECOMPOSITION OF STOCHASTIC FLOWS AND ROTATION MATRIX

2002 ◽  
Vol 02 (01) ◽  
pp. 93-107 ◽  
Author(s):  
PAULO R. C. RUFFINO
Keyword(s):  
Asymptotic Behaviour ◽  
Constant Curvature ◽  
Symmetric Matrix ◽  
Rotation Matrix ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Affine Transformations ◽  
Simply Connected ◽  
Skew Symmetric Matrix

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φt = ξt°Ψt° Θt, where ξt and Ψt have the same properties of Liao's decomposition and (ξt°Ψt) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξt via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

scholarly journals Randers manifolds of positive constant curvature

2003 ◽  
Vol 2003 (18) ◽  
pp. 1155-1165 ◽  
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Complete Riemannian Manifold ◽  
Flag Curvature ◽  
Dimensional Sphere ◽  
Randers Metric ◽  
Constant Flag Curvature ◽  
Simply Connected

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.

scholarly journals OBSERVABILITY OF BAOUENDI–GRUSHIN-TYPE EQUATIONS THROUGH RESOLVENT ESTIMATES

2021 ◽  
pp. 1-39
Author(s):  
Cyril Letrouit ◽  
Chenmin Sun
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Evolution Equations ◽  
Type Equation ◽  
Small Time ◽  
Type Operator ◽  
Resolvent Estimate ◽  
Damped Wave Equation ◽  
Resolvent Estimates ◽  
Speed Of Propagation

Abstract In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure. First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

Fiber structures of special (4 + 3 + 1) warped-like manifolds with Spin(7) holonomy

2014 ◽  
Vol 11 (08) ◽  
pp. 1450076
Author(s):  
Selman Uğuz ◽  
İbrahim Ünal
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Warped Product ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Simply Connected ◽  
Block Diagonal

A generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uğuz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys.61 (2011) 1093–1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F × B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F1 × F2 where Fi's (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the Fi's are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S3 with constant curvature k > 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied.

scholarly journals Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

2006 ◽  
Vol 58 (2) ◽  
pp. 282-311 ◽  
Author(s):  
M. E. Fels ◽  
A. G. Renner
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Isotropy Subgroup ◽  
Algebraic Classification ◽  
Ricci Flat ◽  
Einstein Spaces ◽  
Simply Connected

AbstractA method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

scholarly journals Resolvent Estimates in l_p for Discrete Laplacians on Irregular Meshes and Maximum-norm Stability of Parabolic Finite Difference Schemes

2001 ◽  
Vol 1 (1) ◽  
pp. 3-17 ◽  
Author(s):  
Michel Crouzeix ◽  
Vidar Thomée
Keyword(s):  
Finite Difference ◽  
Piecewise Linear ◽  
Maximum Norm ◽  
Parabolic Problems ◽  
Resolvent Estimate ◽  
Finite Difference Schemes ◽  
Weighted Norm ◽  
Resolvent Estimates ◽  
Lumped Mass ◽  
Mesh Ratio

AbstractIn an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

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