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3
votes
1answer
88 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 ...
1
vote
0answers
71 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: ...
0
votes
0answers
88 views

Categorizing saddle points of real multivariate polynomials

I have a multivariate polynomial function of N variables $f(x_1,x_2,…,x_N) = x_1 x_2 x_3 .. x_N \left( 1 + \sum_i^N (a_i x_i^2 - x_i) \right)$, where $a_i > 0$ are real positive numbers. By ...
4
votes
1answer
375 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 ...
0
votes
1answer
97 views

Continuity of critical points with respect to a parameterisation.

Hello. I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in ...
1
vote
1answer
296 views

How to explain the condition (C) in critical point theory?

Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f. How to see the meaning of " ...
2
votes
0answers
239 views

What is the analog of “monotonic” for scalar functions on surfaces?

"monotonic" is well defined for functions $f(x)$, where e.g. $x\in[0,1]$ and $f(x)\in\mathbb{R}$. The quality I particularly care about is that if $f(x)$ is monotonic then it will not have any local ...
1
vote
1answer
150 views

Critical points in Hilbert space [closed]

Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$. Suppose the restriction of $f$ on $Y$ has a critical point $x_0 \in Y$. Q: Is $x_0$ a critical point of $f$?
3
votes
1answer
445 views

Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset?

Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$. A ...