# Which geometric variational problems admit an entropy identity?

Question. My question in broad terms: when and how can entropy be used in geometric variational contexts to study sequences of critical points, such as the two results described below? [See at the bottom for a more precise version of the question.]

Several branches of mathematics use a concept called entropy, though its definition is notoriously nebulous. My question is about the concept of entropy in geometric analysis, and in variational settings in particular. I am interested in its use to improve the convergence of sequences of critical points. For example:

• Lamm  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  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.

Unfortunately I cannot draw any conclusions from the two examples that go beyond naive observations.

Question. In practical terms, in which geometric-variational settings can one expect to have an entropy identity (in this style) appear? Say a sequence of functionals $$(E_k)$$ is given, depending on parameters $$(\lambda_k)$$ and one wants to study the convergence of a sequence of critical points $$(u_k)$$. Under what hypotheses the triple $$(E_k,\lambda_k,u_k)$$ can an entropy be used, and what conclusions can one hope for?

One last comment: there are other uses of the word (even within geometric analysis): see for instance the Gaussian entropy defined by Colding and Minicozzi . As far as I can tell they have little to do with the listed examples; I am interested in the variational setting above all, and specifically how entropy can be used to study the convergence of sequences of critical points.

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

• It should be noted that a concept of entropy has played an important role in geometric heat flows, notably in Perelman's proof of the Poincare and Thurston conjectures. Before that, Ben Chow introduced it into the study of the Gauss curvature flow. Oct 17 at 16:21
• @DeaneYang That's a fair comment. Unfortunately the title turned out a bit misleading, because the scope of my question is narrower than it would suggest. Do you have a suggestion for an alternative? Oct 17 at 16:40
• The entropy functional also appears prominently in the Lott-Villani-Sturm theory of optimal transport, which you may wish to look into. Oct 17 at 16:40
• The concept of entropy in geometric analysis appears to be just as mysterious as it is in other settings. Oct 17 at 16:43
• @HollisWilliams Ah, interesting - does it have anything in common with the examples I describe? Oct 17 at 16:58