# 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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### Is there a Morse function that does not arise from a minimal-dimensional height function?

Let $M$ be a smooth, finite dimensional manifold of dimension $n$, and let $m$ be the minimal dimension for which $M$ admits a smooth embedding into $\mathbb R^m$.
Question: Does every Morse function ...

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### Is there a canonical "minimal" Morse decomposition?

Classical Morse theory gives a handle decomposition of a finite dimensional manifold $M$ for every choice of Morse function $f: M \to \mathbb R$. Is there always a Morse function that induces a "...

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### Is it likely that gradient flow trajectories of a $G$-invariant function pass through degenerate points?

This question may be posed somewhat vaguely, but I'm interested to actually get an idea of what to expect, so I try to not target it at a specific result.
Assume that $G$ is a compact Lie group, ...

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### Continuity of the set of critical points of a Morse function when the function varies

Let me first give the result I am looking for, then what I found up to now and some related questions.
Definition/Notation.
Denote $Y_k(f)$ the set of critical values $x$ of $f$ such that the index of ...

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### Relation between number of critical points of harmonic functions and number of connected components of the level sets

I am asking what I think is a simple question in the general area of Morse theory, specialized to 2-d and harmonic functions. I'll be specific.
Suppose I have $U(z)$ positive and harmonic for $z\in \...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology

I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...

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### Existence of Morse function on suspension

Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...

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### Questions related to Morse theory

I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood ...

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### Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...

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### Existence of a proper Morse function

I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...

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### Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...

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### A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$

Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...