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Full disclosure, I posed this question on MSE about a week ago, but realized it may be a better fit here after it sat dormant. For completeness, I'll include the full body of the question here:

Let $\boldsymbol{u}:\mathbb R^n\to \mathbb R^n$ be a $C^2$ vector field, representing the velocity of a (steady) fluid flow. If we let $\Phi_t(\boldsymbol x)$ be the flow map for the field $\boldsymbol u$, i.e. $$\partial_t\Phi_t(\boldsymbol x) = \boldsymbol u(\Phi_t(\boldsymbol x)),\quad \Phi_0(\boldsymbol x) = \boldsymbol x$$ for every $\boldsymbol x$, then $(\boldsymbol x,t)\mapsto \Phi_t(\boldsymbol x)$ is $C^2$ and $$\frac{d}{dt} D\Phi_t(\boldsymbol x) = Du(\Phi_t(\boldsymbol x))D\Phi_t(\boldsymbol x).$$ So, by Jacobi's formula, we find that $\det D\Phi_t$ is $C^1$ with $$\frac{d}{dt}\det D\Phi_t = (\det D\Phi_t(\boldsymbol x))(\nabla\cdot \boldsymbol u)(\Phi_t(\boldsymbol x)).$$

Let $\Omega\subseteq \mathbb R^n$ open bounded connected with $C^1$ boundary, so that $\Omega$ is a nice packet of fluid. Set $\Omega_t = \Phi_t[\Omega]$ for $t \in \mathbb R$. Then, for $t > 0$, a change of variables (the details of which I'll spare the reader) yields the Reynolds transport theorem $$\frac{d}{dt}\int_{\Omega_t}\rho(\boldsymbol x, t)\,d\boldsymbol x = \int_{\Omega_t} \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\boldsymbol u)\,d\boldsymbol x$$ for any $\rho: \mathbb R^n\times [0,\infty)\to \mathbb R$ which is (say) jointly $C^1$ in $(\boldsymbol x, t)$ and $C^1$ in $t$ uniformly in $\boldsymbol x$. In particular, plugging in $\rho = 1$, we find $$\frac{d}{dt} \mathcal L^n(\Omega_t) = \int_{\Omega_t}\nabla\cdot\boldsymbol u\,d\boldsymbol x.$$ This little argument proves the perhaps intuitive claim that "the flow of $\boldsymbol u$ preserves volume globally iff $\boldsymbol u$ preserves volume locally, i.e. $\nabla\cdot\boldsymbol u = 0$, $\boldsymbol u$ is incompressible."

A similar argument, made for appropriate vector fields $\boldsymbol F$, yields $$\frac{d}{dt}\int_{\partial \Omega_t} \boldsymbol F(\boldsymbol x, t)\cdot \boldsymbol \nu(\boldsymbol x, t)\,d\mathcal H^{n-1}(\boldsymbol x) = \int_{\partial \Omega_t}\left(\frac{\partial \boldsymbol F}{\partial t}+(\nabla\cdot\boldsymbol F)\boldsymbol u\right)\cdot \boldsymbol \nu\,d\mathcal H^{n-1}(\boldsymbol x)$$ where $\boldsymbol \nu(\boldsymbol x, t)$ is a continuous outward normal to $\partial \Omega_t$ at a point $\boldsymbol x$.

Here's my main question. Try as I might, I cannot figure out how to extend this style of reasoning to understand the quantity $$\frac{d}{dt}\int_{\partial \Omega_t}\rho(\boldsymbol x, t)\,d\mathcal H^{n-1}(\boldsymbol x).$$ In particular, I'm interested in which flows $\boldsymbol u$ preserve surface area. This seems much more restrictive than volume preservation – an informal argument using infinitesimal spheres suggests that $\boldsymbol u$ should probably be incompressible: I find it hard to believe that the volume of (the forward image of) a small ball around a point with nonzero divergence can change while the surface area stays constant without causing some spikes in the boundary (which can't exist because $\partial\Omega_t$ is the image of a $C^1$ manifold under a $C^1$ diffeomorphism.)

I think some argument using the isoperimetric inequality $\mathcal H^{n-1}(\partial \Omega_t) \geq c\mathcal L^n(\Omega_t)^{1-1/n}$ gets us somewhere in this direction. This shows that if the flow of $\boldsymbol u$ preserves surface area, then the volume of $\Omega_t$ is bounded for all $t > 0$ by a (dimensional) constant multiple of the surface area of $\partial \Omega_0$. Maybe then you show that the surface can't collapse in on itself without creating spikes? Unfortunately isoperimetry only goes one way...

However, in 2D any flow of the form $(f(y), 0)$ is incompressible and can stretch surface areas wildly, so surface-area-preservation ought to be much stronger.

A concrete example of a 2D incompressible flow which doesn't preserve surface area is $\boldsymbol u(x,y) = (y,0).$ The flow map induced by $\boldsymbol u$ is $\Phi_t(x,y) = (x+ty,y),$ which stretches a square (perimeter $4$) to a parallelogram with perimeter $2+2\sqrt{1+t^2}$ at time $t$ if I did my computation correctly.

In conversation with some friends, we suspect that this condition implies $\boldsymbol u(\boldsymbol x) = A\boldsymbol x+\boldsymbol b$ for $\boldsymbol b \in \mathbb R^n$ and $A\in \mathbb R^{n\times n}$ antisymmetric, i.e. $A^T = -A$. Indeed, these flows are surface-area-preserving since $\exp(At)$ is orthogonal when $A$ is antisymmetric; they're also incompressible for the same reason. However, we don't have a rigorous proof for general $n$ that these are all of them. Any ideas? An expression for $\frac{d}{dt}\int_{\partial\Omega_t} \rho\,d\mathcal H^{n-1}$ in the same style as our other two derivations would be fantastic, though I'm perfectly happy with resolving the original question in the title, classifying all surface-area-preserving flows as precisely these guys.

I've proven that the flow of a $C^1$ vector field $\boldsymbol u$ is an isometry if and only if the field $\boldsymbol u$ has the form we suspect. So we suspect the following is true, but aren't sure how to prove it: if $\boldsymbol F:\mathbb R^n\to \mathbb R^n$ is a $C^1$ diffeomorphism which preserves the $(n-1)$-dimensional Hausdorff measure of embedded $C^1$ manifolds (with boundary), then $\boldsymbol F$ is an isometry. Any ideas towards proving or disproving this would also be incredibly helpful.

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One can generalize your question: which smooth bijective maps $\Phi$ between regions in $R^3$ preserve surface areas of all surfaces. I claim that only affine orthogonal maps $\Phi(x)=Mx+b$, where $M$ is an orthogonal matrix) do, which implies your conjecture.

First look at the problem on infinitesimal level. Evidently, orthogonal transformations preserve areas. Now if you have any linear transformation $A$, you can apply to it the singular value decomposition $$A=U\Sigma V,$$ where $U,V$ are orthogonal, and $\Sigma$ is diagonal. If $A$ preserves areas then $\Sigma$ must preserve them as well. It easily follows that all diagonal entries are equal to $\pm1$. Indeed, these entries must satisfy three equations $\lambda_j\lambda_k=\pm1,$ for all distinct $j,k$ from $\{1,2,3\}$. These 3 equations are easy to solve and obtain that $\lambda_j=\pm1$. Therefore $A$ is orthogonal.

Now apply this to the derivative $D$ of our map $\Phi$. It follows that the map is conformal (or anti-conformal). By Liouville's theorem, all conformal maps between domains in $R^3$ are Mobius, and since $D$ is also orthogonal, we easily conclude that our map must be an affine orthogonal map. Finally if your map embeds into a flow, than we must have $\det D=1$ (rather than $-1$).

Same proof applies to any dimension $\geq 3$. But it is also true in dimension $2$: a map which preserves lengths is affine orthogonal. The proof in this case is simpler since we don't need Liouville's theorem to conclude that the map is conformal with complex derivative of absolute value $1$. Such maps are complex affine maps.

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  • $\begingroup$ Based on intuition alone, I agree that preserving surface area preserves angles and hence the map would be conformal. I'm not terribly sure how you'd rigorously prove that, though. Also, is $D$ what I called $\boldsymbol F$, or $\Phi_t$? Thanks! $\endgroup$ Commented Sep 2 at 17:46
  • $\begingroup$ @Robert Trosten: My $D$ is what you call $D\Phi_t$, but in my argument $t$ is fixed, so I don't write it. $\endgroup$ Commented Sep 3 at 11:39
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In any Riemannian manifold or even in any manifold with a reversible Finsler metric, the hypersurface area integrand (be it Busemann-Hausdorff or Holmes-Thompson) determines the metric. It follows that area-preserving maps are necessarily isometries.

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  • $\begingroup$ Mea culpa, but would you be able to provide a reference for either of the two area integrands you described? Googling "Busemann-Hausdorff" gets me something abut Busemann functions, which doesn't seem right... $\endgroup$ Commented Sep 2 at 17:44
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    $\begingroup$ See my paper with Thompson, Volumes in Normed and Finsler spaces. The relevant theorems are in Section 6 on the injectivity of the Busemann or Holmes-Thompson area definitions. researchgate.net/publication/… $\endgroup$ Commented Sep 2 at 20:50
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    $\begingroup$ Section 2 of my paper "Rigidity results for geodesically reversible Finsler metrics" is more Finsler (rather than just normed spaces). You'll find it in the ArXiv. $\endgroup$ Commented Sep 2 at 20:53

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