By S. Alinhac
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Additional info for Geometric Analysis of Hyperbolic Differential Equations: An Introduction
Then DL L = 0, showing that an integral curve of L (that is, x˙ = L) is a geodesic. This follows from the symmetry of the Hessian, since for any X DL L, X = − DL ∇u, X = − DX ∇u, L = DX L, L = 12 X L, L = 0. b. Bicharacteristics To the metric g is associated the function p(x, ξ ) = g αβ ξα ξβ . This is a well-defined function on the cotangent space of the manifold M, which is the principal symbol of the wave operator . From a PDE point of view, it is important to consider (null) bicharacteristic curves of starting from a (characteristic) point (x0 , ξ 0 ), which are defined by d α x ≡ x˙ α = ∂ξα p, ξ˙α = −∂α p, x α (0) = x0α , ξα (0) = ξα0 , p(x0 , ξ 0 ) = 0.
Proof First, L, L = 0, since L L, L = 2 DL L, L = 0 and, for s = 0, L, L = gαβ x˙ α x˙ β = 4p(x0 , ξ 0 ) = 0. 24 Computing with frames Next, [L, X] = 0 by construction. Then, for the induced connexion D on the cone, L X, L = DL X, L + X, DL L = [L, X], L + DX L, L = 12 X L, L = 0. Since X, L goes to zero when s goes to zero, X, L = 0 and the orthogonal to L is the tangent plane to the cone. ♦ Consider now a one-parameter family of geodesic cones such that there exists a function u that has the cones of this family as level sets.
3. We use the multiplier X = ea T with a = a(u). This gives additional terms in Qπ: a (u)Q(L, L + L) = 2a (u)(|α|2 + ρ 2 + σ 2 ), and identifies the good components. The corresponding energy density is Q(X, ∂t ) = 12 ea Q(T , aL + a −1 L), justifying the definition of E. 3 (see  for details). 4 Conformal inequalities 1. In the flat case, this quite miraculous inequality (see , ) is obtained using the multiplier K0 = (r 2 + t 2 )∂t + 2rt∂r . Recall that K0 is a timelike conformal Killing field.