证明
Definition 2 ([Spacetime conventions]). A Lorentz metric G has signature…Proposition 36 ([Area envelope and regular patches]). The least perimeter of…观点
Fix a smooth lapse test s compactly supported in the open exterior, and choose a local Gaussian Lorentzian metric inducing g,K at time zero. Its spatial metric and first normal jet are fixed, while its second normal metric jets are free. Choose a two-form whose initial electric and spatial components are c_n E and zero. Its first normal derivatives can be chosen so that both Maxwell equations hold on the initial slice. In general dimension the tangential Maxwell constraints are D_m=0 and α=0 ; the remaining equations prescribe the n(n-1)/2 magnetic and n electric normal evolution components. The induced flux-form variations under normal motion with velocity sN are compactly supported and closed, as follows also from Cartan's formula applied to these initial Maxwell jets. Let S denote the electromagnetic stress tensor with the normalization in Definition 2 ([Spacetime conventions]). It is divergence-free at the initial slice. For each δ>0 , prescribe there the spatial components of Einstein minus S to be T^ (δ) _ ij =((J_m)_i(J_m)_j)/(μ_m+δ). Spatial trace reversal of the second normal metric jets is invertible, so these are legitimate smooth jet prescriptions. The normal and mixed components remain μ_m,J_m by Gauss--Codazzi. The resulting variation has h=2sK . We check that C'→0 uniformly on compact active test sets. The tensors in converge spatially in C^1 . On μ_m>0 this is ordinary smooth convergence; near a zero of μ_m , DEC gives |J_m|≤μ_m and smoothness gives μ_m=∇J_m=0 at that zero. The bounds |T^ (δ) |≤μ_m, |∇T^ (δ) | ≤2|∇J_m|+|μ_m| make the convergence uniform across the zero set. At a positive-density active ray, the limiting spacetime tensor is γ⊗γ/μ_m , where γ(N)=μ_m and γ|_ TΩ =J_m . It is causal and annihilates the null vector N+v . Parallel-transport that null vector along spatial curves. Its contraction with the causal covector is nonnegative and vanishes at the base point; differentiation at the minimum therefore shows that all relevant spatial covariant derivatives, with last argument N+v , vanish. Bianchi and the initial Maxwell equations then give the same limiting vanishing for the normal derivative contracted with N,N+v . Changes of the first argument cost zero, and changes of the second cost zero because it remains null and γ is proportional to its metric dual. At a matter zero, both the tensor and its spatial derivatives vanish by , and the divergence equation again suffices. These observations prove the asserted uniform convergence of C' . Apply to these tests and pass to the limit. They have no boundary or ADM variation, so ∫_ F_* ∫_ ∂^*F str_ T∂^*F K\, A_g\,π(F)=0. At free regular points, Proposition 36 ([Area envelope and regular patches]) provides uniform noncrossing graph patches, and frontiers through a common point have the same tangent plane. Consequently the tangential trace is a single-valued continuous function on each such patch of the integrated surface measure. There is no cancellation between different tangent planes in. A countable cover and Fubini imply the first conclusion, initially almost everywhere and then everywhere on regular sheets by smoothness. At contact, the W^ 2,p graph and the smooth obstacle have the same second jets almost everywhere on their coincidence set. The frontier therefore satisfies H+tr_ tan K=0 almost everywhere throughout the contact graph, including its free portion. Uniform ellipticity first improves its regularity and then bootstraps it to a smooth solution of the expansion equation. The strong comparison principle gives coincidence with S near contact. The set of coincidence is open and closed along the connected S , so the whole S is present. Its area is already the minimum A , leaving no area for another sheet. If there is no contact, the compact frontier is separated from S , and the stated equations hold on all its regular points.