Convexity conditions related with $1/2$ estimate in boundary problems with simple characteristics. II
Masatake Kuranishi · Journal of Differential Geometry · 1972
KURANISHIChoose a submanifold (not necessarily closed) Jί λ of S*E/, which is transversal to C λ and intersects C λ only at (jt°, ζ'O 0 )).Pick a nonzero u in W\x\ ζ λ (x 0 )) = the image of p{(x\ ζ'O 0 )).Then the function viewed as a function on Jί λ is of class C°° and nonnegative, and (JC°, ζ λ (x 0 )) is its isolated zero, i.e., an isolated critical point of f u on Jί λ .Definition 2.3.Assume that the characteristics of A is smooth.We say that a characteristic (jc°, ζ ;We say that the characteristics of A are nondegenerate when each characteristic is so.Since f u on Jί ι takes the minimum value at (JC°, ζ^Oc 0 )), the above condition means that the Hessian of f u on Jf λ at (jc°, ζ λ (x 0 )) is positive definite.In terms of a chart (θ 19, θ k ) of Jί λ with center (JC°, ζ^JC 0 )), this means that the k X kmatrix (d 2 f u /dθ r dθ r ,) (0) is positive definite.If (*°, ζ^Λ: 0 )) is nondegenerate for a choice of a pair of Jf ι and a local trivialization of E, it is also so for any other such choice.We can check this by writing down how f u and its Hessian change when we make a different choice.Note on this connection that a(x°, ζ λ (x 0 )) .pl(x°, ζ λ (x 0 )) = 0.Because of ( 9) and ( 10Hence by ( 11) and ( 15), the nondegeneracy condition means that F λ (x° ;w,χ)\ W λ (x°, ζ λ (x 0 )) is injective for all w € T τ0 N λ and χ J_ ζ^jc 0 ).Thus we have Proposition 2.1.Assume that the characteristics of A are smooth and the projection C λ -> Ό is bijectίve, and further that (JC°, ζ λ (x 0 )) is a nondegenerate characteristic.Then F λ (x°;w, χ), restricted to W(x°,ζ x (x 0 )), is injective for sufficiently small w e N λ and any χ _L ζ λ (x°) provided (w, χ) ψ 0.Lemma 2.8.Under the assumptions in Proposition 2.1, for any ε > 0 we