The tag has no usage guidance.

learn more… | top users | synonyms

2
votes
1answer
170 views

How to obtain the local bound on the length of the Morse function?

This is a follow-up of the question Is there a bound on the length of the longest Morse trajectory?. Consider the following setup. Let $(M,g)$ be a (complete) Riemannian Hilbert manifold and $f$ a ...
2
votes
1answer
326 views

When a Morse function is also a Moment map…

Are there any interesting results, if a Morse function $M \rightarrow \mathbb{R}$ happens to be the moment map, for some $S^1$ action on $M$ (equipped with $\omega$) as well? Thank you very much.
5
votes
4answers
472 views

Is there a bound on the length of the longest Morse trajectory?

Given a compact Riemannian manifold (with a fixed metric) and a Morse function on it (also fixed). Is there a bound (depending on the metric and the Morse function) on the length of the Morse ...
21
votes
2answers
1k views

Graphs, K-theory and combinatorial balls: conjectures

The following conjectures from Kapranov and Saito's Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions aren't as well-known as they aught to be, so I'd like to state ...
0
votes
0answers
271 views

Gradient of the energy functional in $H^{1,2}$-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
1
vote
0answers
143 views

Do higher dimensional maxima of a real valued multivariable function form a cell complex?

Suppose $f: R^n \rightarrow R$ is a positive real valued function. Let $\lambda_1, \ldots , \lambda_i$ be the first $i$ ordered eigenvalues of the Hessian $Hess(f)$. Let $v_1, \ldots, v_i$ be the ...
2
votes
3answers
926 views

Index of a Morse function via the Hessian tensor

For a smooth function $f:M\to \mathbb{R}$ one usually defines the degeneracy and index of a critical point $p\in M$ in terms of the eigenvalues of the Hessian matrix $(\partial^2 f/\partial ...
3
votes
4answers
775 views

literature on geometrical viewpoint on calculus of variations for physics

What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model? ...
6
votes
2answers
2k views

existence of Morse functions satisfying the Palais-Smale condition

Given an arbitrary Hilbert manifold, can on find a complete Riemannian metric and a Morse function satisfying the Palais-Smale condition?
2
votes
2answers
850 views

Morse Theory on Non-smooth Manifolds

Let $X$ be a circle that with one corner (i.e. think of a triangle where we smooth out two of the vertices). Now let us consider the topological torus $M \cong \mathbb{T}^n$ which is the product of ...
23
votes
2answers
2k views

Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon. Does there exist a ...
24
votes
5answers
2k views

Is there a Morse theory proof of the Bruhat decomposition?

Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double ...
5
votes
1answer
828 views

Morse-Bott homology for infinite-dimensional manifolds

Is there any work on Morse-Bott homology for infinite-dimensional manifolds (e.g. Hilbert manifolds). I am particularly interested in the case where we have a locally trivial fiber bundle and the ...
9
votes
6answers
751 views

Is there a Morse theory for sections of bundles or more generally for maps?

This question was prompted by my interpretation of a question by cosmologist Berian James. Background Some cosmologists have suggested using the cosmological dark matter density, which defines a ...
9
votes
6answers
2k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
6
votes
2answers
1k views

Critical points on a fiber bundle

Consider a (smooth) bundle E→_B_, and a (smooth) function f: E → R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting ...