Characterising critical points of
Characterising critical points of
Let
Note that
Question: Does every critical point of
satisfy ?
Of course, by counting degrees of freedom, this doesn't look reasonable.
The converse direction follows from the Euler-Lagrange equation of
where
Thus
Edit 1:
A natural way to produce critical points of a functional is to look at its symmetries.
(In fact I proved that a conformal map
Edit 2:
As I shall prove below (see "Edit 3"), in dimension
First, let's try to answer a more degenerate version of the question (forgetting
Is there a smooth map
, which satisfy
If there is such an
Here
We can try to look first for harmonic counter-examples, but so far I failed doing even that.
Edit 3:
The answer is positive for
, so we need to restrict our attention to .
Indeed, in this case
Since
The Euler-Lagrange equation of
and
(The Euler-Lagrange equation of the Jacobian functional
Expanding equation
We now use the fact the Jacobian functional is a null-Lagrangian, i.e. every smooth map is a critical point, or equivalently
So, the E-L equation
Let
Now we observe that any
K2-Think
The question asks whether every critical point of the functional ( \tilde{E}(f) = \frac{1}{2}\int_{\M} |\bigwedge^2 df|^2 \text{Vol}{\M} ) satisfies ( \delta{\nabla^{\Lambda_2(f^*{\TN})}} (\bigwedge^2 df) = 0 ).
To address this, consider the case where ( \M = \N = \mathbb{R}^3 ) with flat metrics. Let ( f(x,y,z) = (x, y, z^2) ). We analyze ( f ):
- Compute ( \bigwedge^2 df ):...
GPT-5
No, not in general (for d ≥ 3). There are smooth critical points of E that do not satisfy δ∇Λ2(f∗TN)(∧2 df) = 0.
A concrete counterexample in R3
Work on M = N = R3 with the Euclidean metric and consider the “shear” maps f(x, y, z) = (x, y, z + ψ(x, y)), where ψ is a smooth function of (x, y) only.
- Criticality for E
Write Ai = ∂i f, i = 1, 2, 3. Then A1 = (1, 0, ψx), A2 = (0, 1, ψy), A3 = (0...
Claude 3.7 Sonnet
Characterizing Critical Points of the Exterior Power Functional
This is an interesting question about critical points of the functional measuring the "average action on 2D cubes" by a map between Riemannian manifolds.
Understanding the Question
You're studying critical points of:
You've established ...
Claude Opus 4
Looking at this problem, I need to analyze whether every critical point of the functional
Analysis
From the given Euler-Lagrange equation, a map
o3 Pro
No.
Starting with dimension 3 one can find many stationary points of
E(f)=½∫M |∧²df|² dvolM
whose 2–form ∧²df is not co-closed.
The simplest way to see this is to use the well known “hedgehog’’ (or
“Skyrmion’’) type maps. They are spherically–symmetric and can be handled
completely explicitly.
- The set–up...
DeepSeek R1
From the provided context, we know that t...
o4 mini
Here is what one knows and what one does – essentially nothing new beyond what you have already discovered:
-
The true Euler–Lagrange equation of
E(f)=½∫_M |∧²df|²
is not δ(∧²df)=0 but rather the single tensorial equation
h₍f*TN₎ ( tr₍TM₎[ df⊗δ(∧²df) ] ) =0,
where h is your contraction map fTN⊗Λ²(fTN)→f*TN. Equivalently (in local coordinate...
Gemini 2.5 Pro
An analysis of the question leads to the following observations. The question is whether a critical point of the functional
The Euler-Lagrange Equation
The Euler-Lagrange (E-L) equation for
o3
Below is a concise answer that fits into MathOverflow’s “what is the right statement?” spirit.
(Sketches of the routine coordinate computations that back up the two key linear‐algebra
observations are omitted; they can be found in any standard symbolic–algebra package.)
Statement
Let M,N be connected, oriented, d–dimensional ...
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