Question. My question in broad terms: what are the uses of entropy in geometric analysis, particularly in variational contexts to study sequences of functionals 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, and its definition is notoriously nebulous. 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.

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)? Under what hypotheses on a sequence of functionals, 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 examples I listed above; 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.

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.

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 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.

Question. My question in broad terms: what are the uses of entropy in geometric analysis, particularly in variational contexts to study sequences of functionals 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, and its definition is notoriously nebulous. 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.

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)? Under what hypotheses on a sequence of functionals, 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 examples I listed above; 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.