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Concept

N-sphere

Summary

In mathematics, an n-sphere or a hypersphere is a topological space that is homeomorphic to a standard n-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as : S^n = \left{ x \in \mathbb{R}^{n+1} : \left| x \right| = 1 \right} , and an n-sphere of radius r can be defined as : S^n(r) = \left{ x \in \mathbb{R}^{n+1} : \left| x \right| = r \right} . The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally emb

Official source

https://en.wikipedia.org/wiki/N-sphere

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Stability of stationary wave maps from a curved background to a sphere

Sohrab Mirshams Shahshahani

We study time and space equivariant wave maps from

M \times \mathbb{R} \rightarrow S^2

, where

M

is dieomorphic to a two dimensional sphere and admits an action of

SO(2)

by isometries. We assume that metric on

M

can be written as

dr^2+f^2(r)d\theta^2

away from the two xed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where

M

is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

2016

This thesis is a study of wave maps from a curved background to the standard two dimensional sphere S2. The target is always assumed to be embedded in R3 in the standard way. The do- main manifold (the "curved background") will be diffeomorphic to S2 × R, but the Lorentzian metric will not necessarily be the standard one. The current work is based on the following two results by Shatah, Tahvildar-Zadeh [47] and Krieger, Schlag, Tataru [25]. In [47] Shatah and Tahvildar-Zadeh established the existence and stability in the energy norm of a compact family of stationary (satisfying a certain periodicity condition in time) and equiv- ariant (satisfying a certain spatial symmetry) wave maps from S2 × R to S2 equipped with the standard metrics. Stability was proved under perturbations in the same spatial equivariance class. In [25], the authors construct a one parameter family of blow up solutions to the co-rotational wave maps equation from R2+1 to S2 with the standard metrics. Co-rotational means that the wave maps are assumed to have rotation number equal to one. These solutions are obtained as perturbations of the rescaled standard rotational harmonic map 2arctanr from R2 to S2. Here, we prove the conclusions of these papers for different metrics on the domains of wave maps. More specifically, our results can be grouped into two parts: stationary maps and blow ups. In the first part following the method of [25], we construct a one parameter family of co-rotational blow up wave maps from S2 ×R to S2. These solutions can be parameterized by ν ∈ ( 1 , 1]. The metric on the domain sphere is taken to be the standard one. 2 In the second part, we consider wave maps from S2 × R to S2 where the metric on the do- main S2 is now taken to be a general SO(1)−symmetric metric of the form dr2 + f 2(r)dθ2. The special case f (r ) = sin r corresponds to the standard metric considered by Shatah and Tahvildar-Zadeh. We prove the same stability result as in [47] in this general setting.

EPFL2012

We construct blow-up solutions of the energy critical wave map equation on

\mathbb{R}^{2+1}\to \mathcal N

with polynomial blow-up rate (

t^{-1-\nu}

for blow-up at

t=0

) in the case when

\mathcal N

is a surface of revolution. Here we extend the blow-up range found by Carstea (

\nu>\frac 12

) based on the work by Krieger, Schlag and Tataru to

\nu>0

. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere.

2014