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.
