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# Critical point theory under general boundary conditions

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... The geometric situation of tori degenerating to geodesic spheres with shrinking handles corresponds to approaching the boundary of M × BRP 2 , consisting in the vectors of length one in T RP 2 . In order to apply Morse theory to a manifold with boundary (see for instance the classical work of Morse-Van Schaack [31]) it is crucial to understand the normal derivative of the reduced Willmore energy at the boundary of the manifold; this corresponds in our framework to computing the derivative with respect to the Möbius parameter. Such computation is quite delicate since we need sharp estimates and since the torus is degenerating (as it is natural to expect, the computation involves singular integrals), and it will take large part of the present paper (for the final result see Proposition 4.2 and Remark 4.10). ...
... Notice that the factor 1 2 in the definition ofC q is a consequence of the symmetry of the degenerate Clifford torus: indeed for every degenerate Clifford torus there exists a non trivial rotation R ∈ SO(3), R = Id leaving the surface invariant; for more details see Remark 5.1. The existence and generic multiplicity Theorems 1.1-1.2 will then follow from the general results in [31]. ...
... Therefore, by (31), for all η k ≤ η 0 and δ ≤ δ 2 , it follows that ...
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This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the M\"obius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory; to this aim we establish new geometric expansions of the derivative of the Willmore functional on exponentiated small Clifford tori degenerating, under the action of the M\"obius group, to small geodesic spheres with a small handle. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.
... The concept may have been first treated in [22], and though generalizations have since been obtained, see e.g. [21], it is easiest in this paper to work with the ideas and requirements of [22] as they are simpler. ...
... The concept may have been first treated in [22], and though generalizations have since been obtained, see e.g. [21], it is easiest in this paper to work with the ideas and requirements of [22] as they are simpler. ...
... 2) the gradient of f points outwards from the boundary, i.e. its inner product with the outward normal to the boundary is positive. In [22], these conditions are termed "boundary conditions β ". They are presented as a development of "boundary conditions α", which require that f be constant on the boundary. ...
... Since J u is of class C 2 , we have that d is of class C 2 on Λ(R) (14) and t is of class C 1 on Λ(R). We can always suppose ...
... since, if not, we choose c ∈ (0, 1) so that D c := {(t, s) ∈ R 2 : (t, s/c) ∈ D} ⊂ R, and we prove the result with D c in place of D and X c (t, s) := X(t, s/c) in place of X(t, s). 14) It is sufficient to slightly extend Ju and consider d on a small enough tubolar neighborhood of the extension. ...
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We improve an estimate given by Acerbi and Dal Maso in 1994, concerning the area of the graph of a singular map from the disk of R2 into R2, taking only three values, and jumping on three half-lines meeting at the origin in a triple junction.
... Schaak [6] ...
... Remark. The proof is a direct consequence by induction on i E ~ of Theorem 4 of Morse-Van Schaak [6] p. 559 (where the result is proved for a set ~ with just one member). ...
... Since J u is of class C 2 , we have that d is of class C 2 on Λ(R) (14) and t is of class C 1 on Λ(R). We can always suppose ...
... since, if not, we choose c ∈ (0, 1) so that D c := {(t, s) ∈ R 2 : (t, s/c) ∈ D} ⊂ R, and we prove the result with D c in place of D and X c (t, s) := X(t, s/c) in place of X(t, s). 14) It is sufficient to slightly extend Ju and consider d on a small enough tubolar neighborhood of the extension. ...
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In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau's problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of this surface, namely a conformal parametrization defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some result from Morse theory.
... Cette étude a commencé très tôt. Dès 1936, Morse et Van Schaack [11] étudient les modifications homologiques des niveaux des fonctions de Morse sur les variétés à bord. ...
... .11. Torsions de complexes de chaînes qui a défini une notion de torsion pour toute équivalence d'homotopie. ...
Thesis
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Étant donnée une fonction lisse ˜ f définie sur un voisinage de la sphère euclidienne de dimension n dans la boule, peut-on l’étendre en une fonction définie sur la boule bordée par la sphère, de manière à ce que l’extension n’ait aucun point critique ? Cette thèse propose d’étudier cette question, en supposant que la restriction de ˜ f à la sphère, notée f, est Morse. Ce problème a été introduit pour la première fois par Blank et Laudenbach en1970, et a aussi été posé par Arnol’d en 1981. Nous donnons une condition nécessaire d’extension sans points critiques qui s’appuie sur le complexe de Morse de la fonction f, et de la répartition des points critiques de f en deux ensembles : ceux dont la dérivée normale est négative et ceux dont la dérivée normale est positive. Cette condition nécessaire permet alors de donner un cadre algébrique à ce problème venant de la topologie différentielle et s’appuie principalement sur lesgrandes théories de la deuxième moitié du XXème siècle, à savoir celle des cobordismes de Thom,Smale, Milnor etc. Elle permet notamment de donner des conditions nécessaires et suffisantesdans certains cas plus restrictifs, et donne lieu à une condition nécessaire plus faible qui présentel’intérêt d’être calculable.Le point de départ des résultats est celui de Barannikov, qui le premier a traduit le problèmed’extension de fonction avec des conditions de dérivées normales en un problème de chemin defonctions générique qui ne présente pas de singularité globale.
... In this way, one is led to study the finite-dimensional functional Γ = G| Z. In Proposition 3.5 it is shown that, from the informations on Γ at µ = 0 and at infinity, we can apply Morse Theory under general boundary condition, see [18]. Using this technique, we can treat the cases 1) and 2) with the same approach. ...
... For a complete treatment about this topic we refer to [11], where a refinement of the theory in [18] is presented. Let M be a Riemannian manifold, and let f ∈ C 1 (M ). ...
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We show that theN→0 limit of a non-abelian gauge theory is described by an ensemble of self-avoiding random surfaces. This strengthens the connection between gauge and random surface theories.
... In this way, one is led to study the finite-dimensional functional Γ = G| Z . In Proposition 3.5 it is shown that, from the informations on Γ at µ = 0 and at infinity, we can apply Morse Theory under general boundary condition, see [18]. Using this technique, we can treat the cases 1) and 2) with the same approach. ...
... For a complete treatment about this topic we refer to [11], where a refinement of the theory in [18] is presented. Let M be a Riemannian manifold, and let f ∈ C 1 (M ). ...
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We prove the existence of positive solutions for the equation on , where is the Laplace-Beltrami operator on is the critical Sobolev exponent, and is a small parameter. The problem can be reduced to a finite dimensional study which is performed via Morse theory.
... The right branch issued fromv j r−1 inŤ is a new branch whose label isě j r . 20 In Sect. 5, we will prove Proposition 5.4 which is stronger than the present one. ...
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In this article we lay out the details of Fukaya’s A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty$$\end{document}-structure of the Morse complexes of manifold possibly with non-empty boundary. We show that these A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty$$\end{document}-structures are homotopically independent of the made choices. We emphasize the transversality arguments that make some fiber products smooth.
... Following S. Barannikov, we discuss in this section the problem of extending without critical points a germ of function given along the boundary M of a compact (n+1)-dimensional manifold W , M = ∂W . This setting was already considered in 1934 by Morse-van Schaack [6] where Morse inequalities have been formulated and proven for manifolds with non-empty boundary. Notice that generically a function F : W → R is a Morse function whose critical points lie in the interior of W and whose restriction to the boundary is Morse. ...
Article
Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given.
... It can be shown directly that every Morse function f : M → R without critical points represents the trivial element of C n . Namely, a nullcobordism F : W → R 2 of f can be obtained by restricting the map id condition for the extendability of a non-singular ∂M -germ [g : [0, ε) × ∂M → R] to a non-singular Morse function M → R, we recover the Morse equalities χ + [g] ≡ χ(M ) (mod 2) (for n even) and χ + [g] = χ(M ) (for n odd) due to Morse and van Schaack[14, Theorem 10]. ...
Article
By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all nondegenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine’s cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki–Yamamoto’s cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.
... Starting with ω we use (1) and (2) to find sufficiently many θ j so that (θ 0 : · · · : θ n+1 ), essentially, embeds W \S into CP n+1 . Then, we decrease n as much as we can by a repeated use of a Morse-Whitney-Sard lemma [20,31,26] which implies that if ϕ : M −→ M ′ is a smooth map between Riemannian manifolds and dim R M < dim R M ′ , ϕ (M) has measure 0. such that each α j α λ(j) where α j = ∂η j is a coordinate for R in V j . Hence, Ψ = (α 0 : · · · : α m ) is an immersion of W such that Ψ (z) = Ψ (z ′ ) implies z = z ′ when (z, z ′ ) ∈ V = ∪ 3 j m V j × V j . ...
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In this paper we give results about projective embeddings of Riemann surfaces, smooth or nodal, which we apply to the inverse Dirichlet-to-Neumann problem and to the inversion of a Riemann-Klein theorem. To produce useful embeddings, we adapt a technique of Bishop in the open bordered case and use Runge type harmonic approximation theorem in the compact case.
... (b) According to Lemma 4.2-(b), if (39) holds then ∂ ν Γ(p) = 0 at any critical point of Γ| ∂Z . Hence Γ satisfies the general boundary conditions on ∂Z, see [23]. Moreover, setting ...
... Originally, the Morse theory of smooth functions and their gradient flows on compact manifolds X with boundary has been studied (under general boundary conditions) in the papers of Morse (1929) and of Morse and Van Schaack (1934). See Barannikov (1994) for the case X being a ball. ...
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... Each surgery adds an additional critical point inside the manifold. The surgery Figure 1.Surgery on a strong Morse function eliminating all inward points was used in [4]. It is easy to obtain the classical Morse inequalities by combining this surgery with the results of parts (0), (1) and (3) of Proposition 1.1. ...
Preprint
Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper.
... In this section, we recall results of Borodzik, Nemethi and Ranicki [4] who, among other things, study the topological changes of the level sets (with boundary) of the extension when one goes through a critical point on the boundary. Morse and Van Schaak [18] already knew about Morse inequalities for Morse functions on manifold with boundary. The following results will only be used in the next section. ...
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In this article, we are interested in the problem of extending the germ of a smooth function $\tilde{f}$ defined along the standard sphere of dimension $n$ to a function defined on the ball which has no critical points. The article gives a necessary condition using the Morse chain complex associated to the function $f$, restriction of $\tilde{f}$ to the sphere $\mathbb{S}^n$, which is assumed to be a Morse function.
... We briefly recall the construction from the Milnor's book "Morse theory" [2] of a CW-complex associated with the function f . Note that Morse theory was essentially developed long time before CW-complexes were defined, see [1]. ...
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Consider the set of all rectangular $n \times m$ matrices with entries in a field. Recall that unitriangular group $T_n$ consists of upper triangular matrices with 1's on the diagonal. The product $T_n \times T_m$ naturally acts on the aforementioned set: $X \mapsto AXB^{-1}$. Our first observation is that each orbit of this action contains a unique matrix which has at most one non-zero entry in each row and in each column. Thus these non-zero numbers and their positions are invariants of a matrix under this action. This is a variation of a classical Bruhat decomposition for $GL$. When applied in the setting of Morse theory, this linear algebraic construction leads to invariants of a strong Morse function $f$. Namely, positions of non-zero entries correspond to the well-known Barannikov decomposition (also known as persistent homology) of $f$. The novelty is the values themselves, which correspond to numbers, carried by Barannikov pairs (also known as bars in the barcode). Considering further a complex, constructed from a strong Morse function, we interpret the product of all the numbers as a torsion of chain complex.
... of critical points. Morse (see [3]) has distinguished two types of critical points, based on the fact that if h : M —> R has a single critical point of index À corresponding to the critical value c, then rH p (M t ) behaves in one of two ways as t increases from c — e to c + e: either it increases by 1 in dimension A, or else it decreases by 1 in dimension À — 1 (while in either case it remains unchanged in the other dimensions). Morse speaks thus of critical points of increasing and decreasing type. ...
Article
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1. Einleitung In der vorliegenden Arbeit werden zwei Haupts~itze der MorseTheorie f/Jr berandete Mannigfaltigkeiten bewiesen. Einige Untersuchungen zu Mannigfaltigkeiten mit R~indern findet man bereits bei Morse und van Schaak [8] sowie in der Monographie von Morse und Cairns [7]. Im Gegensatz zu den genannten Arbeiten sollen hier auch Mannigfaltigkeiten mit Kanten zugelassen werden. Der Begriff des kritischen Punkts ist fiir Funktionen auf berandeten Mannigfaltigkeiten geeignet zu fassen; denn Punkte, die ffir den Abstieg zu niedrigeren Niveaus kritisch sind, brauchen fiir den Aufstieg keineswegs kritisch zu sein. Die Situation veranschaulichen wir an Hand eines Beispiels, das sich - abgesehen von dem Rand - in [5] findet. Die Abb. 1 zeigt einen 2-dimensionalen Torus, fiir den V die Tangentialebene in einem Punkte p' ist. Durch eine auf V senkrecht stehende Ebene IV, W~p', wird die Torusfl~iche in zwei Teile geschnitten. Wir betrachten als berandete Mannigfaltigkeit M die abgeschlossene Teilfl~iche, die nicht p' enth~ilt. Sei f:M~IR die H6he fiber IX, und f/Jr reelles a sei Ma:= {x~M; f (x) < a}. Dann erkennt man: (1) Wenn a< f(p) gilt, ist M a leer.
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By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine's cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki-Yamamoto's cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.
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