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### Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post
Normal form for a holomorphic Morse function
Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...

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862 views

### Two-convexity ⇒ Lefschetz?

Assume that
$\Omega$ is an open simply connected set in $\mathbb R^n$
(two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$.
Is it true that any component of ...

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2k views

### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem.
Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$.
Then for $\ell$ a linear form on $E$, we have a ...

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197 views

### Intersection of plus/minus cells in Bialynicki-Birula decomposition

Let $X$ be a projective variety endowed with an algebraic $\mathbb{C}^*$-action. Assume the fixed point set $W$ is discrete. Then we have two cell decompositions of $X$:
$X = \bigsqcup_{w\in W} C_w$ ...

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167 views

### Is there a general theory of Sard measures?

Let $f:M \to N$ be a smooth map of smooth (second countable) manifolds. The set $C_f \subset M$ of critical points of $f$ is defined to be the set of all $m \in M$ such that the differential $df : ...

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511 views

### Good introduction to Morse-Novikov theory?

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...

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129 views

### Perturbation of Morse functions at critical points leaving stable manifolds invariant

Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).
Is there a way ...

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409 views

### Topological version of two results in smooth Morse theory

Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).
It is well known ...

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161 views

### Generators of local homology groups of an isolated critical point

This is a basic Morse theory question:
Let $M$ be a smooth manifold, and $f:M\to\mathbb{R}$ a smooth function with an isolated critical point $x$. Set $c:=f(x)$. The local homology of $f$ is the ...

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287 views

### Is the Hessian almost everywhere nondegenerate?

Let $M$ be a complete Riemannian manifold. For a fixed point $p$ in $M$, the Riemannian distance to $p$ is denoted by $d_p$. Fix a strongly convex geodesic ball $B(o,R)$ in $M$ and some disjoint ...

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135 views

### complex Morse function on a four-manifold

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...

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119 views

### Detecting Non-Transversality

Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad ...

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248 views

### Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them.
Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...

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142 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 ...

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91 views

### Application of Morse theory to second order systems

Hello
I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems )
Someone can help me with a pdf or a book which has these applications ?
...

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88 views

### Can Morse trajectories break if their first derivative is uniformly bounded?

Consider a compact Riemannian manifold endowed with a Morse function f. Fix two critical points x and y of f. Whenever you have a sequence $u_k$ of Morse trajectories connecting x and y it might ...

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216 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 functional ...