On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

Abstract

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

Holographic reconstruction and renormalization in asymptotically Ricci-flat spacetimes

“…An asymptotically hyperboloidal initial data set [17,[25][26][27] (see also [28,29]) is defined as any initial data set (Σ,…”

“…Recall from (3.21) that our defining function ρ = z/β. By factorising the latter, our spacetime solution reads as: 26) with N (0) := N (t)/β(t). The metricG of the conformal embedding is given byG µν = (z/β) 2 G µν and the expectation value of the operator dual to g (0) is therefore obtained as: 24…”

“…The expression for E follows from an explicit computation of (4.28) for each value of d and has been verified up to d = 6. 26 As an observation, by using the definitions (4.27) and (4.28), equation (4.30) can be rewritten as: 31) and which represents the holographic Callan-Symanzik equation for a anomaly-free CFT deformed by a source N (0) for a relevant scalar operator of dimension one. The Weyl anomaly of each field theory can therefore also be seen as that created by a non-vanishing vacuum expectation value of such an operator.…”

Construction of Hyperboloidal Initial Data

“…Then a solution u = o(x −1 ) is of the form u = 1 + u 1 x + u 2 x 2 + u 3,1 x 3 ln x + u 3 x 3 + higher order terms, where u 3,1 = cf 3 for some explicit constant c. This example reflects the fact, cf. [6,4,5] that solutions to degenerate elliptic systems are in general nonsmooth at ∂M , instead the general form of the solution has a polylogarithmic expansion of the form u = u i,j x i ln j x. In case the RHS and the coefficients are smooth, the logarithm terms in u appear first at the critical exponent α + .…”

“…In order to get a regular evolution at I , these data must be regular up to ∂M . It was shown by the author, Chrusciel and Friedrich [6,4,3,5] that under certain conditions on the boundary geometry of M , the Cauchy data for the conformally regular field equations, are smooth up to ∂M .…”