Questions tagged [morse-theory]
In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.
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Integral equivariant formality for Hamiltonian T-actions
What is the simplest example of a compact symplectic manifold $M$ with Hamiltonian $T$-action for which the integral $T$-equivariant cohomology is not formal
i.e. $$H_T^*(M,\mathbb{Z}) \not \cong H^*(...
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How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
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Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
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Generalize the conception of 'stable' solution and 'stable outside a compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold
I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold.
I'm reading $\Delta u +e^u=0$...
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Perturbation of Morse function
The following question is probably classic in Morse theory, so a reference to an existing result should be sufficient. I don't know much about Morse theory and I am dealing with the following ...
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What is the infinite Morse index solution?
I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered
$$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$...
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Morse theory for manifolds with boundary
I need a reference to some basic facts about Morse theory on manifolds with boundary.
Namely, if a critical point lies on the boundary, then the gradient of function might be nonzero and it brings ...
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Dynamical analogue of Morse theory
Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{...
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Integrability (and hence regularity) of $\alpha$-harmonic maps
To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
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Critical points on a fiber bundle
Consider a (smooth) bundle $E\to B$, and a (smooth) function $f: E\to\mathbf{R}$ on the total space. Then it makes sense to talk about the derivatives of $f$ along the fibers. Let $C$ be the ...
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Is squared geodesic distance a Morse-Bott function on simply-connected Hadamard manifolds?
Let $M$ be a simply connected Hadamard manifold. That is, $M$ is a complete Riemannian manifold with sectional curvature bounded above by 0. Let $f(x,y)=\frac{1}{2}d^2(x,y)$, where $x,y \in M$, and $d(...
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A closed leaf with two different index with respect to two different Riemannian metrics
Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:
Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...
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Homogenization of Morse-Bott functions
Let $M$ be a compact manifold of dimension $n$.
A smooth function $f:M \to \mathbb{R}$ is called Morse-Bott if the set critical points of $f$ is a disjoint union of compact submanifolds $C_1,\ldots,...
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Reference for Morse-Bott vector fields
I'm looking for a reference for the following result:
Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ ...
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Is Morse theory local?
I am currently trying to use Morse theory in my work. Say I have $X$ smooth compact manifold and $f : X \to \mathbb{R}$ a smooth function. The classical result I know is : when $f$ has non-degenerate ...
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Wrinkling smooth functions
I am interested in applying a result from the work by Eliashberg and Mishachev on wrinkling. Namely, in their first paper on wrinkling, they prove Theorem 1.6 B (Theorem 1.6 A is a non-parameterized ...
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Handle attachment information from Morse function and triangulation
First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$.
For simplicity, let's restrict for now to the ...
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Morse theory for isotopies of codimension 1 submanifolds and generalized bigon moves
I seem to have managed to convince myself of the validity of a certain result. If it does indeed hold (modulo non-catastrophic adjustments) I could really use a reference. Otherwise, I would also be ...
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Sard's theorem and Cantor set
Sard's famous theorem asserts that
Theorem. The set of critical values of a smooth function from a manifold to another has Lebesgue measure $0$.
I am asking for the curiosity that is it possible to ...
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Understanding dimension of gradient flow trees for product on Morse complex
I am trying to square my intuition with the facts and hope this question is not too vague.
I am reading this paper which describes the $A_\infty$-category of Morse functions on a manifold $M$ of ...
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Transformation of Morse function in $\mathbb{CP}^n$ trough symplectomorphism
I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.
Consider $(\mathbb{CP}^n,\omega)$ ...
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Visualizing a normalized hyper-elliptic curve as a honey-flow on a multi-doughnut
Trying to visualize in 3D the normalized model $y^2=f(x)$ of a hyper-elliptic curve AND the degree two ramified cover $(X,Y)\mapsto X$, $\mathbb{C}\times \mathbb{C}\to \mathbb{C}$. Is this projection ...
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Morse theory for compact sets bounded by hypersurfaces in euclidian space
I am having trouble understanding precisely how some part of Morse Theory works.
More precisely, take $X$ to be a compact set of $\mathbb{R}^d$ such that $\partial X$ (topological boundary) is a ...
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How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
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What is the definition of a height function for a subsurface?
What is the definition of a height function for a subsurface? Is a height function exactly a Morse function?
In the paper:
"On the Teichmüller tower of mapping class
groups
By Allen Hatcher at ...
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The difference between a handle decomposition and a CW decomposition
Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it ...
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The norm-squared of a moment map behaves like a Morse-Bott function
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Let $<.,>$ denote a $G$-invariant inner product on $\mathfrak{g}$.
Let $(M,\omega)$ be a symplectic compact manifold endowed with ...
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Stable sets for gradient flow of functions with singularities
Let $f: \mathbb{R}^n \to \mathbb{R}$ be a real-analytic function, and let $F_t$ denote the gradient flow of $f$ with respect to some background metric. Suppose that $df = 0$ at a point $p$. In the ...
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The negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical point?
Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth ...
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Morse index in PDEs
I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came ...
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Perturbation of vector fields in Morse Homology
Recently I have been reading on Morse Homology. Suppose we have a compact manifold $M$ and a smooth function $f:M \rightarrow \mathbb{R}$ and a Morse vector field $X$ such that we can do Morse ...
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Diagrams for critical points [closed]
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) pages 13 and 15 we have :
for case "d&...
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configurations of three saddles on one level [duplicate]
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...
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Why do we have sixteen possible configurations of three saddles on one level?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...
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Morse functions inducing Heegaard diagrams
Let $(\Sigma, \alpha, \beta)$ be a Heegaard diagram for a 3-manifold $M$, corresponding to a Heegaard splitting $M = H_1 \cup_\Sigma H_2$. There may be many self-indexing Morse functions $f: M \to \...
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What are good Morse Theory lecture notes and books?
Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these ...
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Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
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A question about Marino–Prodi perturbation
In this paper N. Ghoussoub,
the author claims the following version of Marino–Prodi perturbation, that is :
Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...
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Existence of connections in a vector bundle whose parallel transport preserves a function on a total space
Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space ...
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Stratification of smooth maps from R^n to R?
I'm interested in stratifications of smooth maps $\mathbb{R}^n\to\mathbb{R}$ (or more generally of any $n$-manifold $M^n\to\mathbb{R}$). The codimension 0 stratum should be Morse functions, and the ...
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Is an $L^p$-sphere in Sobolev space $H_2^{s}(\Omega)$ a Hilbert manifold?
For a bounded smooth domain $\Omega$, let $H_2^{s}(\Omega)$ be the usual Sobolev space on $\Omega$.
Define $A:=\{f\in H_2^{s}(\Omega)| \lVert f\rVert_{L^p(\Omega)}=1\}$ where $2<p<2_{s}^*$.
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Good introduction to Morse-Novikov theory?
Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
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What are your common strategies/remedies when your new theory/idea stuck in most cases?
Sorry if this is not a suitable post for MO.
Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the ...
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Genericity of an induced projection map
I am cross-posting a question asked on Math Stackexchange that has not been answered, in which I am still interested in.
Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\...
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Operations via Morse Theory
I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)...
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Building a manifold from a CW complex inductively
Given a finite dimensional finite $CW$ complex $X$ of dimension $d$, I want to build a compact manifold $M$ (with least dimension possible) with boundary with the property that,
$M$ has the same ...
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Proper Morse function on open set
Let $M$ be a compact submanifold with boundary of $\mathbb{R}^n$ of dimension $n$. Let $f:M \to \mathbb{R}$ be a Morse function. Then $f$ is proper. Let $N:=M-bd(M)$. How can I get a proper Morse ...
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Dismissing pseudoholomorphic curves in embedded contact homology
In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...
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Morse function on the Sphere base on functions on the disc
Lets consider a Morse function $f:\mathbb{S}^2\rightarrow \mathbb{R}$ such that it has two maximal points, one minumun and one saddle at $c$. Notice that $f^{-1}(-\infty,c)$ is topologically a disk.
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Homological stability of Chow varieties
Given a connected component $C$ of the degree $d$ Chow variety of $r$ cycles on $C_{d,r}(X)$ ($X$ is smooth projective variety over $\mathbb{C}$), let $C'$ be another connected component of $C_{d',r}(...
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