Skip to main content
MathOverflow

Questions tagged [morse-theory]

Ask Question

In differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold.

277 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
3 votes
0 answers
187 views

I have a closed (compact without boundary) manifold M and a compact Lie group G that acts on it. I want to understand the topology of $M/G$, at least compute its singular homology groups. The action ...
fnfgjvtuldiyswuqauexqqhwdswjib's user avatar
  • 31
0 votes
0 answers
57 views

Consider the following complex oscillatory integral on a neighborhood $U\subset \mathbb{R}^n$: \begin{equation} \mathscr{I}(a)=\int_{U}e^{a S(x)}\varphi(x)\, d^nx \end{equation} The functions $S(x)$ ...
color's user avatar
  • 139
3 votes
1 answer
195 views

Let $f$ be a Morse-Smale function on a smooth manifold $M$ without boundary. Suppose all critical points have distinct critical values. Let $s < t$ be two regular values of $f$. Define $M^s = \{x \...
siuc's user avatar
  • 51
1 vote
1 answer
135 views

Consider the motion space of a linkage of 3 rods of equal length = 1 with the first vertex anchored to the origin (see attached image). Take the standard negative height function $$f(\theta_i)=-\sum_{...
sean's user avatar
  • 73
9 votes
0 answers
183 views

Consider the configuration space of distinct points in a sphere, $$C_k S^n = \{ p \in (S^n)^k : p_i \neq p_j \ \forall i \neq j \}.$$ Here, $S^n = \{ p \in \mathbb R^{n+1} : |p|=1 \}$ is the unit ...
Ryan Budney's user avatar
  • 45.5k
8 votes
2 answers
271 views

$\newcommand\O{O}%In case \\\$\operatorname O\\\$ might be acceptable, just change this to `\DeclareMathOperator\O{O}` \DeclareMathOperator\tr{tr}$Let $\O_n$ be the orthogonal group of $n \times n$ ...
Ryan Budney's user avatar
  • 45.5k
1 vote
0 answers
139 views

Let $f:Y \to X$ be a surjective projective morphism between complex varieties (not necessarily smooth) with connected fibers. Let $x\in X$ be a closed point. I want to know when there exists an ...
Li Yutong's user avatar
  • 3,584
4 votes
1 answer
202 views

Let $M$ be a compact smooth manifold and $f\in\mathscr{C}^\infty(M)$ be a Morse function with distinct critical values. Such functions are exactly stable smooth functions on $M$, that is, those smooth ...
Enrico Savi's user avatar
  • 43
1 vote
1 answer
158 views

Suppose $H$ is a Hilbert space, $O\subset H$ an open subset, $M\subset O$ a smooth submnifold of $H$, so the Riemannian metric on $M$ is just the inner product of $H$. Now, consider a smooth ...
KingKermit's user avatar
  • 11
2 votes
0 answers
162 views

The following theorem can be thought of as a generalization of the Morse lemma for vector-valued functions. Let $\varphi : \mathbb{R}^n \to \mathbb{R}^k$ be a smooth map such that: $\varphi(0) = 0$, ...
Gordhob Brain's user avatar
  • 369
2 votes
0 answers
136 views

Let $M$ be a Riemannian manifold. If $P \to M$ is a smooth prinicpal $G$-bundle, then let $\mathcal{C}(P;G)$ denote the moduli space of $G$-connections on $P \to M$. Well studied examples of ...
user82261's user avatar
  • 449
4 votes
0 answers
145 views

If $(f_t)_{t\in [0,1]}$ is a smooth family of Morse functions on a closed manifold $M$, does there necessarily exist a smooth family of Riemannian metrics $(g_t)_{t\in[0,1]}$ so that $(f_t,g_t)$ is a ...
Aakash Parikh's user avatar
  • 41
3 votes
0 answers
184 views

I have been reading about Bott's iteration theorem on the index of iterates of closed geodesics (see, for example, here p.172 and here -- p.224). Here is a summary of the setup and my question. Let $M$...
Talant Talipov's user avatar
  • 51
0 votes
0 answers
72 views

Let $(M,g)$ be a compact 2-dimensional Riemannian manifold other than the sphere. Let $\gamma:S^1\to M$ be a contractible loop on $M$ (which need not be simple). $E(\gamma):=\frac{1}{2}\int |\gamma'|^...
Jianqiao Shang's user avatar
  • 83
5 votes
1 answer
423 views

Consider a manifold $M$ with a base point $*$. Denote by $\Omega^\textrm{nd}_*M$ the space of all smooth loops based at $*$ with non-degenerate parametrization, i.e. the tangent vector with respect to ...
Grisha Taroyan's user avatar
  • 1,536
2 votes
0 answers
91 views

Given a domain $D\subset \mathbb{R}^{2n}$, possibly with corners, such that $\partial D=H^{-1}(1)$ for certain Hamiltonian $H:\mathbb{R}^{2n}\rightarrow \mathbb{R}$. Let's assume that I have some ...
kvicente's user avatar
  • 211
0 votes
0 answers
113 views

Are two Morse functions $H$ and $\widetilde{H}$ defined on the same surface $S$ having the same critical points (with the same index) equivalent? In the sense that there exists diffeomorphisms $h\...
Odylo Abdalla Costa's user avatar
  • 151
5 votes
1 answer
266 views

Let $U\subset{\mathbb R}^n$ be a bounded open domain with smooth boundary. I assume that $U$ is diffeomorphic to a ball. You may think of $L=\Delta$ and $U$ is the unit ball. Let $L=\operatorname{div}(...
Denis Serre's user avatar
  • 53.1k
5 votes
1 answer
469 views

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 ...
Nate River's user avatar
  • 9,980
3 votes
0 answers
245 views

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 "...
Nate River's user avatar
  • 9,980
2 votes
0 answers
116 views

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, ...
Whatsumitzu's user avatar
  • 137
2 votes
0 answers
113 views

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 ...
Chev's user avatar
  • 63
3 votes
0 answers
114 views

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^*(...
onefishtwofish's user avatar
  • 1,315
1 vote
0 answers
124 views

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 ...
Azur's user avatar
  • 111
2 votes
1 answer
176 views

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 $\Delta u +e^u=0$...
Elio Li's user avatar
  • 1,041
4 votes
1 answer
341 views

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.$...
Elio Li's user avatar
  • 1,041
5 votes
1 answer
983 views

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 ...
Anton Petrunin's user avatar
  • 47.1k
4 votes
1 answer
316 views

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^{...
Ali Taghavi's user avatar
  • 400
1 vote
0 answers
96 views

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^...
Wai's user avatar
  • 219
1 vote
1 answer
297 views

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(...
Spencer Kraisler's user avatar
  • 528
2 votes
0 answers
106 views

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,...
Sergiy Maksymenko's user avatar
  • 859
2 votes
0 answers
162 views

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 $X$ ...
Paul's user avatar
  • 1,419
2 votes
0 answers
199 views

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 ...
Taraellum's user avatar
  • 93
3 votes
1 answer
346 views

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 ...
rab's user avatar
  • 191
0 votes
0 answers
187 views

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 $M$, ...
Ali Taghavi's user avatar
  • 400
5 votes
1 answer
424 views

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 ...
Mathguest's user avatar
  • 1,691
3 votes
0 answers
215 views

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 ...
asymmetriad's user avatar
  • 151
2 votes
0 answers
203 views

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 ...
mcwiggler's user avatar
  • 121
2 votes
0 answers
243 views

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 $(\mathbb{CP}^n,\omega)$ ...
kvicente's user avatar
  • 211
2 votes
1 answer
197 views

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 ...
Taraellum's user avatar
  • 93
2 votes
2 answers
574 views

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 ...
Mira's user avatar
  • 159
6 votes
1 answer
484 views

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 ...
Mira's user avatar
  • 159
5 votes
0 answers
258 views

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 ...
Nick Salter's user avatar
  • 2,860
6 votes
2 answers
2k views

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 ...
Cat's user avatar
  • 781
3 votes
1 answer
211 views

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 ...
Someone's user avatar
  • 811
6 votes
2 answers
709 views

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 ...
Mira's user avatar
  • 159
1 vote
0 answers
104 views

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&...
1200785626's user avatar
  • 204
0 votes
0 answers
50 views

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 ...
Usa's user avatar
  • 119
0 votes
1 answer
291 views

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 ...
Usa's user avatar
  • 119
3 votes
1 answer
228 views

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 \...
Pepijn's user avatar
  • 31

1
2 3 4 5 6