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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 [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.

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 [3]. 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.

[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.

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  • $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Oct 17 at 16:21
  • $\begingroup$ @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? $\endgroup$
    – Leo Moos
    Oct 17 at 16:40
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    $\begingroup$ The entropy functional also appears prominently in the Lott-Villani-Sturm theory of optimal transport, which you may wish to look into. $\endgroup$ Oct 17 at 16:40
  • $\begingroup$ The concept of entropy in geometric analysis appears to be just as mysterious as it is in other settings. $\endgroup$
    – Deane Yang
    Oct 17 at 16:43
  • $\begingroup$ @HollisWilliams Ah, interesting - does it have anything in common with the examples I describe? $\endgroup$
    – Leo Moos
    Oct 17 at 16:58
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In Kahler geometry, there is a long-standing open problem of determining which manifolds admit metrics of constant scalar curvature (cscK). In particular, there is a conjecture due to Yau, Tian and Donaldson which states that for particular class of Kahler manifolds, the existence of a cscK metric is equivalent to an algebro-geometric condition known as K-polystability.

It is known that K-polystability is necessary for the existence of a cscK metric, and that they are equivalent of Fano manifolds. However, the general conjecture remains open because it is very difficult to use algebro-geometric data to find a cscK metric, which involves proving a priori estimates for a particular fourth-order PDE.

Recently, Chen and Cheng [1] made a breakthrough and showed that it is possible to obtain all the a priori bounds needed for the existence problem if you can bound a single quantity, which is the entropy of the cscK metric with respect to some background metric. Using this, they were able to prove quite a few conjectures in this subject and relate the existence problem to stability conditions in the space of Kahler metrics. My understanding is that they are not yet able to solve the full YTD conjecture, but that this was a major breakthrough in the area. As a disclaimer, I am not an expert on cscK metrics, so please let me know if I have any misconceptions here.

[1] Chen, Xiuxiong; Cheng, Jingrui, On the constant scalar curvature Kähler metrics. I: A priori estimates, ZBL07397078.

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  • $\begingroup$ Shoot, I'm sorry... I appreciate your comments, but it's not quite what I meant to ask. I am not so much interested in examples---I am sure there are many---, but rather a conceptual point of view. Moreover, I am specifically interested in entropy identities used to control the convergence of a sequence of functionals, like in the two examples. I will update the question to make that clearer. $\endgroup$
    – Leo Moos
    Oct 18 at 8:15

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