87 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

With respect to the first part of your question: No, a function with two global minima does not necessarily have an additional critical point. A counterexample is $$ f(x, y) = (x^2-1)^2 + (e^y - x^2)^...
Martin R's user avatar
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32 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

$ \def\norm#1{\lVert#1\rVert} $The answer to the question as stated is no as Martin showed, but is yes if we add the condition that $f(x)→∞$ as $\norm{x}→∞$. Martin's example pushes the saddle point '...
user21820's user avatar
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14 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

In 2019, I posted an answer to a relevant question. See: Can a multivariate function only have local minimum?, and Can a smooth function with compact sublevel sets only admit local minimizers? In [1], ...
River Li's user avatar
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12 votes
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Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions?

This is an almost complete answer. There is currently one small gap, with regards to non-degeneracy, but I am relatively certain that this can be fixed: tl;dr If there are $n$ masses, then in general, ...
mlk's user avatar
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11 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Morse theory says that every Morse function $f$ (that is all critical points were non-degenerate and distinct critical points take distinct critical values) satisfies $$\#\min+\#\max-\#\mathrm{saddle}=...
C.F.G's user avatar
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9 votes
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Understanding Morse homology with the example of a circle

Let's solve the issues one by one. Let's also use the case of a deformed sphere with the height function as the Morse function, which is similar to the circle but has more structure so that we ...
Aloizio Macedo's user avatar
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6 votes

Geometric Intuition of Eigenvalues of Hessian Matrix

The comments to your question cover it pretty well, but the key piece that I think you might be missing is that (assuming appropriate conditions on the derivatives) the Hessian matrix $H$ is symmetric....
amd's user avatar
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6 votes
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Extensions of vector fields and the Hessian

An extension of $v \in T_pM$ to a vector field $V$ is just any $V \in \Gamma(TM)$ such that $V_p = v$. There are many extensions of a given vector, which is why we must show that the result is well-...
Anthony Carapetis's user avatar
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Poincare Duality in Morse Homology

If $f$ is a Morse function on the closed manifold $V$, then so is $-f$. The index $k$ critical points of $f$ are index $(n-k)$ critical points of $-f$, and the boundary map $$\partial_k: C_k(V,f) \to ...
marston morse's user avatar
5 votes
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Can a real-valued function on the sphere have exactly 2 critical points which are not antipodal?

Take the function $f(x_1, \dots, x_{d}) = x_{d}$ defined on $\mathbb S^{d-1}$. Let $\phi : \mathbb S^{d-1 }\to \mathbb S^{d-1}$ be a diffeomorphism such that $\phi (0,\cdots, 0,1) = (0,\cdots, 0,1)$ ...
Arctic Char's user avatar
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5 votes

Morse lemma for holomorphic functions

The following proof comes from the book "The monodromy group" by Henryk Żołądek (the author says that the proof was suggested by T. Maszczyk). First you need the lemma of Hadamard (see wikipedia for ...
Nicolas Hemelsoet's user avatar
5 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Suppose we have a compact orientable manifold $M$ with boundary $\partial M$. Suppose that we have a Morse function $f$ whose gradient flow is transverse to the boundary $\partial M$. Then the ...
Thomas Rot's user avatar
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Question about the proof of Reeb's theorem in Milnor's Morse Theory

Let us look closer at the implications the normal form for $f$ given by Morse Lemma. Centered at $p$, being the minimum point (in particular a local minimum), there is a chart $(y,U)$ with $y(r)=\big(...
Laz's user avatar
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What is the definition of Index of a point or vector field?

For every isolated zero of a vector field, the nearby points are nonzero, therefore for a small sphere around the zero, we may assign a unit vector. This is therefore a map from $S^n\to S^n.$ The ...
ziggurism's user avatar
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Reference request for Morse theory with complex valued functions

The correct holomorphic analogue of Morse functions on compact complex manifolds is a Lefschetz pencil (see also and references therein); see also here. A LP is a holomorphic map $f: M\to {\mathbb C} ...
Moishe Kohan's user avatar
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Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Probably you can write down an explicit contraction, but recall that $\pi_{k-1}(\Omega_{p,p}(M))\cong \pi_{k}(M)$. This comes from the path space fibration $\Omega_{p,p}(M)\rightarrow P(M)\rightarrow ...
Thomas Rot's user avatar
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Potential proof for the Slice-Ribbon conjecture (may be wrong).

When thinking about this problem notice the following, there are plenty of proper smooth embeddings of disks which cannot be approximated by ribbon embeddings. Just take any disk and connect sum it ...
PVAL-inactive's user avatar
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Prerequisites for discrete Morse theory

Does one "need" to know Morse theory to learn discrete Morse theory? No, go for it. But if you lose motivation along the way, wondering what it's all about, here's some further thoughts. Does it "...
Lee Mosher's user avatar
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4 votes
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Homotopy equivalence onto special fiber

As mentioned in the comment, Clemens proved the case for normal crossing divisors. The general case will follow from this result, which I'll provide proof later. Theorem. (Clemens, 1977) If $X$ is a ...
AG learner's user avatar
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4 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

We want to explain the intuition in the OP : at the level where the lakes join, there exists a critical point. Let $M$ be a topological space that is normal, locally path connected and path-connected (...
orangeskid's user avatar
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4 votes
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A Morse Function with Minimum Number of Critical Points

Permit me to talk about the Morse complex over $\mathbb {Z}$. For $\mathbb{Z}_2$ a similar argument works. A function whose Morse complex has trivial boundary maps is called a perfect Morse function. ...
Thomas Rot's user avatar
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4 votes
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Morse lemma via Moser's trick

Since $df(0) = 0$, given a vector $v$ we can write $$ df(x)(v) = \big<G(x)v,x\big>, $$ where $$ G(x) = \int_0^1 \mathrm{Hess}\;f(sx)\;ds. $$ In fact, $$ df(x)(v) = \int_0^1 \frac{d}{ds}df(sx)(v)...
João Caminada's user avatar
4 votes
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A closed manifold has closed geodesics of at most countably many lengths

It's been a while (well, 20 years) so I won't be able to fill in all the details, but the following should allow you to figure this out. Maybe someone who is still working in the area can provide an ...
Thomas's user avatar
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4 votes

Is any vector field without periodic orbits a gradient field?

This is false. Any function on a closed manifold must have critical points (for example corresponding to the maximum or the minimum), hence the corresponding gradient vector field must have equilibria....
Thomas Rot's user avatar
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4 votes
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Intuitive explanation of singular homology

Singular homology groups are the most fundamental algebraic invariant of topological spaces. A traditional but terrible metaphor claims that the singular homology groups measure the presence of holes ...
Kevin Arlin's user avatar
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4 votes

"Simple to state, but difficult to solve" problems which require analyzing topology of simplicial complexes?

A lot of work on the evasiveness conjecture uses topology of simplicial complexes. Here is a bit of an overview of the evasiveness conjecture: Suppose we have a graph on $n$ vertices and we want to ...
David E Speyer's user avatar
3 votes

Morse functions with minimal number critical points

Your first question: no. A Morse function on $RP^1 \cong S^1$ has at least two critical points (a maximum and minimum). More generally, the Morse inequalities tell us that the number of critical ...
Alvin Jin's user avatar
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3 votes

Can a real-valued function on the sphere have exactly 2 critical points which are not antipodal?

I think that $f_{A,B}(C) = \frac{d(A,C)^2}{d(A,C)^2+d(B,C)^2}$ should do it. This may help with the intuition: Consider the Earth as $S^{d-1}$ with $d=3$. Let $f$ be a function that, given a point on ...
Acccumulation's user avatar
3 votes
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Milnor - Morse Theory, proof of Morse's lemma

The matrix $M=\{H_{i,j}(0)\}_{r\le i,j\le n}$ being symmetric and nondegenerate means it defines an indefinite inner product on $\mathbb{R}^{n-r+1}$. Hence there exists some $y\in \mathbb{R}^{n-r+1}$ ...
Florian's user avatar
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