By Abbas Bahri
This publication offers a brand new entrance of analysis in conformal geometry, on sign-changing Yamabe-type difficulties and get in touch with shape geometry particularly. New floor is damaged with the institution of a Morse lemma at infinity for sign-changing Yamabe-type difficulties. This kinfolk of difficulties, considered out of succeed in many years in the past, turns into a relations of difficulties that are studied: the publication lays the root for a software of analysis during this direction.In touch shape geometry, a cousin of symplectic geometry, the authors turn out a primary results of compactness in a variational challenge on Legrendrian curves, which permits one to outline a homology linked to a touch constitution and a vector box of its kernel on a three-d manifold. The homology is invariant less than deformation of the touch shape, and will be learn on a sub-Morse complicated of the Morse complicated of the variational challenge outfitted with the periodic orbits of the Reeb vector-field. This e-book introduces, for this reason, a pragmatic instrument within the box, and this homology turns into computable.
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Of Lemma 13 and H¯ older. If = k and λ ≥ λj , 1/2 ω |ωk | ≤ 4 ω Bj 4 Bj ωk2 1/2 ω4 . Lemma 15 Ωci ≤ ω 4 |ωk |δi + ω 4 |ωk |δj + Bj (|ωk∞ | + |ωs∞ |)εks + ε3ks εsi O s=i ω 4 |ωk | λj O(εij ) Ωi 5/2 k + 0(εij ) ( Max εjs O(ε2ik εij ) + εij + k ∞ (|ωj∞ | + |ωm |)εjm + εjm ). 5/2 Proof. We use 5 of Lemma 13 for the first term, 7 and 9 of Lemma 13 for the second term and 2 of Lemma 13 for the third term. Lemma 16 Ωci Proof. |v|5 δi + Ωi |v|5 δj + O(εij λj ) Straightforward, via H¯ older. Bj |v|5 = O √ εij |∇v|2 5/2 .
We are left with ω 5 for = i or ω 4 ωk or the like. Q∗ of such terms involve first these terms themselves. 2]. In addition, there are the projection terms related to A−1 . Those corresponding to e , with = i, are again controlled by the previous lemma. The term corresponding to ei is typically: .. −1 A − ω 4 ωk L−1 em ei . .. i Here, we need to understand more the matrix A and its inverse. 3 The matrix A A can be written as B+C where C is an almost diagonal matrix which separates the block i from the block j: () 0 0 B = 0 () 0 .
On an estimate on |h Next, using the Green’s function of an annulus-type domain, a pointwise ¯ i (Lemma 12). estimate is derived on h The estimate is complex because it involves (via the right hand side of the equation satisfied by v¯) an enormous amount of terms and expressions ¯ j etc. related to Ωi but also Ωj , for j = i, (U Ωi )c . It ties v¯i , ¯hi with v¯j , h We need to estimate carefully all of these expressions. This is what we complete in Lemmas 13—30. We then are in position to derive an estimate (not yet optimal) on |¯ vi |H01 ¯ and |hi |∞ (Lemmas 31—34).