**Question.** My question in broad terms: what is the definition of entropy in geometric analysis, particularly in variational contexts such as the one described below? [See at the bottom for a more precise version of the question.]

Several branches of mathematics use a concept called *entropy*, in seemingly not always directly related ways. It originates in thermodynamics, but is also prominent in statistics and probability theory—see the [question](https://mathoverflow.net/questions/146463/what-is-entropy-really) for interpretations in this context.

My question is about the concept of entropy in geometric analysis, and specifically in variational settings. For example:
- Lamm [1] studies $\alpha$-harmonic maps $u_{\alpha_k} \in C^\infty(M,N)$ imposes an entropy condition of the form \begin{equation} \liminf_{k \to \infty} (\alpha_k - 1) \int_M \log( 1 + \lvert \nabla u_{\alpha_k} \rvert^2) (1 + \lvert \nabla u_{\alpha_k} \rvert^2)^{\alpha_k} = 0, \end{equation}
as $k \to \infty$ and $\alpha_k \to 1$.
This is used in a blow-up analysis to prove an energy identity.
- Riviere [2] assumes an entropy condition of the form \begin{equation} \sigma_k^2 \int_\Sigma (1 + \lvert A_k \rvert^2)^p \mathrm{d} \, \mathrm{vol}_{g_k} \in o\Big(-\frac{1}{\log \sigma_k}\Big) \end{equation} as $k \to \infty$ and $\sigma_k \to 0$, where the $\Phi_k$ are $W^{2,2p}$-immersions of $\Sigma^2$ into a manifold $N^n \subset \mathbf{R}^Q$. Again this is used to study the convergence, and prove that the $\Phi_k$ converge to a conformal harmonic map.


A few elementary observations can be made: 1. it seems to be defined asymptotically, for a sequence of critical points, 2. it introduces a logarithm somewhere, 3. it seems to improve weak convergence to a stronger form. I am unable to draw any conclusions beyond these naive comments.

**Question.** In practical terms, in which geometric-variational settings can one expect to have an entropy condition appear? Under what hypotheses on a sequence of functionals, and what conclusions can one hope for?

One last comment: there is another use of the word (within geometric analysis) in dynamic settings, where time-dependent equations are studied: see for example the Gaussian entropy defined by Colding and Minicozzi [3]. As far as I can tell they have little to do with one another; I am interested in the variational meaning above all.



[1] T. Lamm. Energy identity for approximations of harmonic maps from surfaces. *Trans. Am. Math. Soc.*, 362:4077-4097, 2010.   
[2] T. Riviere. A viscosity method in the min-max theory of minimal surfaces. *Pub. math. IHES*, **126**, 177-246 (2017).  
[3] T. Colding and W. Minicozzi. Generic mean curvature flow I: generic singularities, *Ann. of Math.* (2) **175** (2012), no. 2, 755-833.