# Convex Integration Theory: Solutions to the h-principle in by David Spring

By David Spring

This ebook offers a accomplished examine of convex integration concept in immersion-theoretic topology. Convex integration concept, constructed initially via M. Gromov, presents common topological equipment for fixing the *h*-principle for a wide selection of difficulties in differential geometry and topology, with purposes additionally to PDE conception and to optimum regulate thought. notwithstanding topological in nature, the speculation is predicated on an exact analytical approximation outcome for greater order derivatives of services, proved via M. Gromov. This ebook is the 1st to offer an exacting checklist and exposition of all the uncomplicated suggestions and technical result of convex integration idea in better order jet areas, together with the speculation of iterated convex hull extensions and the speculation of relative *h*-principles. A moment function of the ebook is its certain presentation of purposes of the overall concept to themes in symplectic topology, divergence loose vector fields on 3-manifolds, isometric immersions, completely genuine embeddings, underdetermined non-linear structures of PDEs, the relief theorem in optimum regulate conception, in addition to purposes to the normal immersion-theoretical subject matters akin to immersions, submersions, *k*-mersions and loose maps.

The booklet may still end up invaluable to graduate scholars and to researchers in topology, PDE concept and optimum regulate idea who desire to comprehend the *h*-principle and the way it may be utilized to unravel difficulties of their respective disciplines.

------ reports

The first 8 chapters of Spring’s monograph include an in depth exposition of convex integration idea for open and considerable relatives with exact proofs that have been usually passed over in Gromov’s e-book. (…) Spring’s ebook makes no try to comprise all themes from convex integration thought or to discover all the gemstones in Gromov’s basic account, however it will still (or accurately accordingly) take its position as a typical reference for the idea subsequent to Gromov’s towering monograph and will end up fundamental for someone wishing to profit concerning the concept in a extra systematic approach.

- Mathematical Reviews

This quantity presents a accomplished examine of convex integration thought. (…) We instructed the publication warmly to all drawn to differential topology, symplectic topology and optimum keep an eye on theory.

- Matematica

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**Extra info for Convex Integration Theory: Solutions to the h-principle in geometry and topology**

**Example text**

Evidently the pair (h, H) is a C-structure over Op b with respect to f, β. 2 26 II. 2, one patches together the local C-structures constructed above over Op b for all b ∈ B. The inductive step for this process is as follows. 5. Let K, L ⊂ B be closed and let (g0 , G0 ), (g1 , G1 ) be C-structures over Op K, Op L respectively with respect to f, β. There is a C-structure (h, H) over Op (K ∪ L) with respect to f, β such that (h, H) = (g0 , G0 ) over (a smaller) Op1 K and (h, H) = (g1 , G1 ) over Op (L \ Op (K ∩ L)).

27) Fs − Fs < δ/2, provided > 0 is suﬃciently small. 23) the image of F , lies in N (δ/2). 27), the image of the homotopy F over Op K lies in N (δ) ⊂ N , and hence F : [0, 1] → Γ(R), provided > 0 is suﬃciently small. Evidently, the pair (f , F ) satisﬁes conclusion (iii) of the theorem, except possibly the requirement that the homotopy F be constant on Op K. To satisfy this additional property, note that, over Op K, F0 = β, F1 = ∂t f = β and that, ﬁberwise over b ∈ Op K, the image of F lies in a δ-ball in Rb , centered at β(b).

Our proof procedure (adapted from Spring [37]) constructs a holonomic section inductively over a countable, locally ﬁnite covering by charts of the base manifold, at the nth stage of which one constructs a section which is holonomic in the nth chart and which coincides suitably on the overlap with the sections constructed during previous steps of the induction. Analytically, the distinguishing feature of spaces of 1-jets is that locally, in coordinates (u1 , u2 , . . , un ) on the base manifold, the 1-jet of a function is expressed in terms of the “pure derivatives” ∂/∂ui , 1 ≤ i ≤ n.