All Questions
Tagged with critical-point-theory dg.differential-geometry
10 questions
40
votes
3
answers
3k
views
Height function on 2-torus with only 3 critical points
It is well-known that a Morse function on $T^2$ has at least $4$ critical points, but also that there exist functions $f\colon T^2\to\mathbb R$ with only 3 critical points (the least possible number ...
- 5,891
5
votes
1
answer
147
views
Regularity requirements for Sard's Theorem
The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$.
Question. Is it ...
- 16.2k
5
votes
1
answer
686
views
Failure of Palais-Smale Condition C and the Mini-Max Principle
To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition ...
- 17.5k
5
votes
0
answers
290
views
Pullback of Morse form satisfies Palais Smale
Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector ...
3
votes
1
answer
387
views
Does this squared distance functional have a unique critical point on geodesically convex manifolds?
Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow \...
- 2,286
2
votes
1
answer
309
views
Bijection of critical points on two manifolds
Suppose that $f$ and $g$ are two smooth functions defined on $R^n$. Assume that $(a-\epsilon, b+\epsilon)$ contains no critical point of $g$. Then $g^{-1}[a,b]$ it homomorphic to $g^{-1}(a)\times [a,b]...
- 21
2
votes
1
answer
134
views
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 ...
- 93
2
votes
0
answers
293
views
Measure of the Attractor of Critical Points of a Manifold
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
- 109
1
vote
0
answers
80
views
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&...
- 204
1
vote
0
answers
117
views
what are definitions of born or die (birth-death point) and crossing point?
in this paper we have :
A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston
...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t}(...
- 119