Exponential map (Lie theory) | EPFL Graph Search

In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects. Definitions Let G be a Lie group and \mathfrak g be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map :\exp\colon \mathfra

RENORMALIZATION AND BLOW-UP FOR WAVE MAPS FROM S-2 x R TO S-2

Sohrab Mirshams Shahshahani

We construct a one parameter family of finite time blow-ups to the co-rotational wave maps problem from S-2 x R to S-2, parameterized by nu is an element of (1/2, 1]. The longitudinal function u(t, alpha) which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from R-2 to S-2. The domain of this harmonic map is identified with a neighborhood of the north pole in the domain S-2 via the exponential coordinates (alpha, theta). In these coordinates u(t, alpha) = Q(lambda(t)alpha) + R(t, alpha), where Q(r) = 2arctan r is the standard co-rotational harmonic map to the sphere, lambda(t) = t(-1-nu), and R(t, alpha) is the error with local energy going to zero as t -> 0. Blow-up will occur at (t, alpha) = (0, 0) due to energy concentration, and up to this point the solution will have regularity H1+nu-.

American Mathematical Society2016

Renormalization and blow up for wave maps from $S^2 \times \mathbb{R}$ to $S^2$

Sohrab Mirshams Shahshahani

We construct a one parameter family of nite time blow ups to the co-rotational wave maps problem from

S^2\times R

S^2

parameterized by

\nu \epsilon (\frac{1}{2},1]

. The longitudinal function

u(t,\alpha)

which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from

\mathbb{R}^2

S^2

. The domain of this harmonic map is identied with a neighborhood of the north pole in the domain

S^2

via the exponential coordinates

(\alpha ,\theta)

. In these coordinates

u(t,\alpha) = Q(\lambda(t)\alpha) + R(t,\alpha)

, where

Q(r) = 2\,arctan\,r

is the standard co-rotational harmonic map to the sphere,

\lambda (t) = t^{-1-\nu}

, and

R(t,\alpha)

is the error with local energy going to zero as t ! 0: Blow up will occur at

(t,\alpha) = (0,0)

due to energy concentration, and up to this point the solution will have regularity

H^{1+\nu-}

2016

Renormalization and blow up for wave maps from $S^2 \times \mathbb{R}$ to $S^2$

Sohrab Mirshams Shahshahani

We construct a one parameter family of nite time blow ups to the co-rotational wave maps problem from

S^2\times R

S^2

parameterized by

\nu \epsilon (\frac{1}{2},1]

. The longitudinal function

u(t,\alpha)

which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from

\mathbb{R}^2

S^2

. The domain of this harmonic map is identied with a neighborhood of the north pole in the domain

S^2

via the exponential coordinates

(\alpha ,\theta)

. In these coordinates

u(t,\alpha) = Q(\lambda(t)\alpha) + R(t,\alpha)

, where

Q(r) = 2\,arctan\,r

is the standard co-rotational harmonic map to the sphere,

\lambda (t) = t^{-1-\nu}

, and

R(t,\alpha)

is the error with local energy going to zero as t ! 0: Blow up will occur at

(t,\alpha) = (0,0)

due to energy concentration, and up to this point the solution will have regularity

H^{1+\nu-}

2016