Prove
Proposition 3 ([Cap regularity]). Fix -1/4<b_*≤0 , and assume. Suppose that,…Proposition 2 ([Patch obstruction]). The metrics in U admitting no admissible…Proposition
Put Y=_y . The definitions and the Darboux equation give H(Y,D)=0, H(D,D)=KE/H_ yy , H(D,_x)=KE/H_ yy . Thus H(D,·)=O(K) on bounded coordinate vectors. The commutation formula for a covariant Hessian gives (∇_DH)(Y,D)=(∇_YH)(D,D)+O(K), because in dimension two the curvature tensor is a bounded tensor multiple of K , and the commutator contracts it against dz . Differentiate H(Y,D)=0 . The term containing ∇_DY is O(K) , and ∇_DD=Γ^i_ DD _i-(Dq)Y . Hence the left side of equals H_ yy (Dq-qΓ^x_ DD -Γ^y_ DD )+O(K). On the right, differentiating H(D,D)=KE/H_ yy gives K_yE/H_ yy +O(K) ; the two connection terms are O(K) . Division by H_ yy , followed by, proves.