Deformation Theorem Research Articles

The deformation theorem for flat chains

The deformation theorem for flat chains

Disjunctive Optimization: Critical Point Theory

In this paper, we introduce the concepts of (nondegenerate) stationary points and stationary index for disjunctive optimization problems. Two basic theorems from Morse theory, which imply the validity of the (standard) Morse relations, are proved. The first one is a deformation theorem which applies outside the stationary point set. The second one is a cell-attachment theorem which applies at nondegenerate stationary points. The dimension of the cell to be attached equals the stationary index. Here, the stationary index depends on both the restricted Hessian of the Lagrangian and the set of active inequality constraints. In standard optimization problems, the latter contribution vanishes.

A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities

We apply the Tian-Todorov method, proving the Bogomolov smoothness theorem (for deformations of compact Kähler manifolds) to deformations of the regular part of a Stein space with a finite number of isolated singular points. By the argument based on the Hodge structure on a strongly pseudo-convex Kähler domain or on a punctured Kähler space, we obtain an unobstructed subspace of the infinitesimal deformation space.

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑Cn be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ¦D∖Z are either Hirzebruch surfaces (projective bundles overP1), Hopf surfaces (elliptic bundles overP1), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.

Global surjectivity of submersions via contractibility of the fibers

We give a sufficient condition for a C 1 {C^1} submersion F : X → Y F:X \to Y , X X and Y Y real Banach spaces, to be surjective with contractible fibers F − 1 ( y ) {F^{ - 1}}(y) . Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard’s criterion when Y ≃ X Y \simeq X and F F is a local diffeomorphism, and the Palais-Smale condition when Y = R Y = \mathbb {R} . These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.

A Note on the Analogue of the Bogomolov Type Theorem on Deformations of CR-Structures

AbstractLet (M, °T″) be a compact strongly pseudo-convex CR-manifold with trivial canonical line bundle. Then, in [A-M2], a weak version of the Bogomolov type theorem for deformations of CR-structures was shown by an analogy of the Tian- Todorov method. In this paper, we show that: in the very strict sense, there is a counterexample.

Generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces

We denote by M(k) the moduli space of all isomorphic classes of H-stable rank 2 vector bundles with fixed determinant bundle D and second Chern class k. It is well known that M(k) is a quasi projective variety, and is not empty if k is sufficiently large (see [G] and [M]). The local structure of M(k) is described precisely by the Kuranishi deformation theorem in the following way. Suppose that V~M(k) and let r H 1 (Endo(V))~ HZ(Endo(V)) be the Kuranishi map, then the stalk of the structure sheaf (gM~) at V is naturally isomorphic to the germs of holomorphic functions at 0~H'(Endo(V)) divided by the ideal generated by the components of ~. In particular, if HZ(Endo(V))=O, then M(k) is smooth at V, and the tangent space T(M(k))v is identified with H 1 (Endo(V)). The Hirzebruch-Riemann-Roch theorem gives the dimension formula

The palais-smale condition versus coercivity

possesses a convergent subsequence; and cp is said to satisfy (PS) if ir satisfies (PS),, for every c/E IR. Recently, among other results in critical point theory, Shujie [I I] showed that if a C’ functional p: X + K’ is bounded from below and satisfies the condition (PS) then p is coercive. Shujie’s proof uses a “gradient flow” approach, through the so-called “deformation theorem” (cf. [3, IO]) and, for that, he needs the notion of a pseudo-gradient vector field tjassociated with the functional cp (whose existence is guaranteed for C’ functionals by Palais [8]). In this paper we present some new results which relate the Palais-Smale condition and the notion of coercivity, and are based on the well-known variational principle due to Ekeland [5, 61. In particular, a new proof of the above-mentioned result of Shujie is given. It should be pointed out that, throughout the paper, the given functional p could be assumed to be only Gateaux differentiable, rather than C’. And, in addition to being conceptually simpler, this approach could be used in more general situations where the functional is not even differentiable (cf. [4]). The strong form of Ekeland’s variational principle, to be repeatedly used is given in the following theorem.

On the Problem of Torsion in Finite Elasticity: Part II, Some General Remarks in the Case of Uniform Torsion

ABSTRACT The problem of finite elastic torsion for beams of a generic solid cross-section is considered. A general theory and two approximate theories are formulated by applying Reissner's variational theorem for finite elastic deformations. Introduction of different assumptions with respect to the stress and displacement fields yields a unified approach to the problem of finite uniform elastic torsion. In such a way, the formulations developed by Seth, Villaggio, and Kappus appear as attempts to solve the problem with a different degree of approximation. Finally, a comparison between the two approximate nonlinear theories and St. Venant's linear theory is performed for beams with rectangular cross-sections.

On the Problem of Torsion in Finite Elasticity:Part I, The General Formulation in the Case of Torsion with Variable Twist

ABSTRACT The problem of nonuniform torsion of beams with solid cross-sections is examined in the finite displacement theory of elasticity. Two approximate theories are formulated by applying Reissner's theorem for finite elastic deformations. A perturbation solution scheme is presented, to study results implied by the two approximate formulations.

Note on Betti's Theorem for Infinitesimal Deformations Superimposed on Finite Deformation of a Plane Curved Beam

ABSTRACT Betti's theorem implies variational principles, without which the convergence of many numerical techniques would be open to question. Even though Betti's theorem applies to exact large-deformation three-dimensional elastic analysis, the same may not be true with respect to approximate analyses based on truncated expansions.

Deformations of complex supermanifolds

The supermanifold analogue of the Kodaira-Nirenberg-Spencer existence theorem for deformations of complex structures is given. It is shown that every complex supermanifold is a deformation of a vector bundle.

Deformation theorems for analytic functions in annuli and discs

Let O be an annulus or a disc in the complex plane and let f be an analytic function on O. A lower bound for the area of the set f(O) is obtained in terms of the length of the curve f(O), where C is a certain circle concentric with O. The inequality is shown to be sharp and the cases of equality are determined. We indicate proofs which lie entirely within the domain of undergraduate mathematics.

A uniqueness theorem for deformations of Fuchsian groups

A uniqueness theorem for deformations of Fuchsian groups

Deformation Theory and the Tame Fundamental Group

Let U be a curve of genus g with $n + 1$ points deleted, defined over an algebraically closed field of characteristic $p \geqslant 0$. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite $p’$-groups on $n + 2g$ generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin’s Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.

Local deformations of isolated singularities associated with negative line bundles over abelian varieties

Let V be an analytic space with an isolated singularity p. In [1] M. Kuranishi approached the problem of deformations of isolated singularities (c.f. [2] and [3]) as follows; Let M be a real hypersurface in the complex manifold V − {p}. Then one has the induced CR-structure °T″(M) on M by the inclusion map i: M→ V − {p} (c.f. Def. 1.6). Then deformations of the isolated singularity (V, p) give rise to ones of the induced CR-structure °T″(M). He established in §9 in [1] the universality theorem for deformations of the induced CR-structure °T″(M)9 when M is compact strongly pseudo-convex (Def. 1.5) of dim M ≧ 5. Form this theorem we can know CR-structures on M which appear in deformations of °T″(M).

A deformation theorem for the Kobayashi metric

Let M 0 {M_0} be a compact hyperbolic complex manifold. It is shown that the infinitesimal Kobayashi metric is upper semicontinuous in a C ∞ {C^\infty } deformation parameter t ∈ U ⊆ R k t \in U \subseteq {R^k} . This is accomplished by proving deformation theorems for holomorphic maps.

Deformation of homeomorphisms on stratified sets

T H E O E M 0. The topological group H(X) of homeomorphisms of a finite simplicial complex X onto itself is locally contractible. This result does not extend to ENR's (euclidean neighborhood retracts). To see this let a space X be obtained from S 3 = 3 u o o by crushing to a point each of a sequence of mutually disjoint wild non-cellular arcs in 3 :,41, a 2 , A 3 . . . . such that each A,, n~> 1, is a copy of the same wild arc A in the unit ball in 3 translated by the vector (4n, 0, 0). This X is an ENR; indeed X x is homeomorphic to S a x = 4 0 by a result of Andrews and Curtis [4]. Clearly this compactum admits self-homeomorphisms h : X ~ X arbitrarily near the identity which nontrivially permute the images o fA 1, A2, A3, .... But no such h is isotopic to the identity because these are isolated points at which Y fails to be a manifold. (See also the fish skeleton of w The treatment of non-manifolds rests roughly speaking on a method for deforming homeomorphisms on R x cX, cY being the open cone on X, once one is given such a method on +1 xX-. Then the proof proceeds by induction on the depth of X. Here Xis regarded as a stratified set, and depth is the greatest difference of dimensions of nonempty strata of X. Stratified sets are vital to the proof because their open subsets are themselves stratified sets, and often of a lesser depth. Thus it will only clarify matters to deal from the outset with suitable stratified sets. I take this opportunity to introduce classes of pleasant stratified sets that may come to be the topological analogues of polyhedra in the piecewise-linear realm or of Thom's stratified sets in the differentiable realm. This technique of proof almost automatically provides strong relative and respectful deformation theorems (w 4.3, w 5.10), which a counterexample (w 2.3.1) suggests are

Complementary Energy Method for Finite Deformations

A principle of stationary complementary energy is developed for statically loaded structures with finite displacements, rotations, and strains, made of material with linear or nonlinear elasticity. The principle is based on a vectorial definition of complementary work analogous to the vectorial definition of ordinary work. It is used to solve two illustrative truss problems, including one in which even the infinitesimal deformations cannot be determined by the conventional complementary energy method. The principle is shown to include the conventional complementary energy theorem for infinitesimal deformations as a special case.

Complementary Energy Theorem for Symmetric Shells

A variational theorem is presented for the stresses and meridional slopes in axially symmetric thin shells with large strains, displacements, and slope changes, made of material having linear or nonlinear elasticity. This theorem represents a generalization of the stationary complementary energy theorem for infinitesimal deformations. The shell may have orthotropy consistent with axial symmetry and its elastic constants may vary meridionally. The following loadings are considered: On the shell middle surface, distributed radial and axial forces of prescribed magnitude per unit undeformed middle-surface area plus radial centrifugal forces; on the shell boundaries, prescribed radial force of displacement, prescribed axial force of displacement, and prescribed meridional bending moment or rotation. The theorem represents a particular application of a general stationary complementary energy theorem developed by the writer previously. In the present paper, the shell theorem is proved independently of the earlier paper by means of the calculus of variations. As a further check, the theorem is shown to reduce to the conventional form when the deformations are infinitesimal.