A Yamabe-type problem on smooth metric measure spaces. (English) Zbl 1334.53031
The following is taken from the introduction of the article:
The article under review “describes and partially solves a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman’s \(\nu\)-entropy.”
“The Yamabe constant and Perelman’s \(\nu\)-entropy are related to sharp Sobolev-type inequalities on Euclidean space, with the Yamabe constant recovering the best constant for the Sobolev inequality and the \(\nu\)-entropy recovering the best constant for the logarithmic Sobolev inequality.”
The author of the article introduced in [Calc. Var. Partial Differ. Equ. 48, No. 3–4, 507–526 (2013; Zbl 1275.49081)] a “one-parameter family of geometric invariants which interpolate between the Yamabe constant and the \(\nu\)-entropy”. Those invariants are called weighted Yamabe constants.
“The purpose of the article is to study to what extend these invariants interpolate between the Yamabe constant and the \(\nu\)-entropy, focusing on issues related to the problem of finding minimizers of the weighted Yamabe quotients.”
The following theorems are shown:
“Let \((M^n,g,e^{-\phi} \mathrm{dvol},m)\) be a compact smooth metric measure space. Then \[ \Lambda[g,e^{-\phi}\mathrm{dvol},m] \leq \Lambda[\mathbb{R}^n,\mathrm{d}x^2,\mathrm{dvol},m] = \Lambda_{m,n}. \] Moreover, if the above inequality is strict, then there exists a positive function \(w \in C^\infty(M)\) such that \[ \mathcal{Q}(w) = \Lambda[g,e^{-\phi}\mathrm{dvol},m]." \] Here, \(\mathcal{Q}\) is the weighted Yamabe quotient, \(\Lambda[g,e^{-\phi}\mathrm{dvol},m]\) denotes the weighted Yamabe constant and \(\Lambda_{m,n}\) is a constant depending on \(m\) and \(n\).
Let \((M^n,g,e^{-\phi}\mathrm{dvol},m)\) be a compact smooth metric measure space such that \(m \in \mathbb{N} \cup \{0,\infty\}\). If \[ \Lambda[g,e^{-\phi}\mathrm{dvol},m] = \Lambda[\mathbb{R}^n,\mathrm{d}x^2,\mathrm{dvol},m], \] then \(m \in \{0,1\}\) and \((M^n,g,e^{-\phi}\mathrm{dvol},m)\) is conformally equivalent to \((S^n,g_0,\mathrm{dvol},m)\) for a metric \(g_0\) of constant sectional curvature. In particular, there exists a positive function \(w \in C^\infty(M)\) such that \[ \mathcal{Q}(w) = \Lambda[g,e^{-\phi}\mathrm{dvol},m]. \]
It is shown that “There does not exist a minimizer for the weighted Yamabe constant of \((S^n,g_0,1^{1/2}\mathrm{dvol})\)”.
The article under review “describes and partially solves a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman’s \(\nu\)-entropy.”
“The Yamabe constant and Perelman’s \(\nu\)-entropy are related to sharp Sobolev-type inequalities on Euclidean space, with the Yamabe constant recovering the best constant for the Sobolev inequality and the \(\nu\)-entropy recovering the best constant for the logarithmic Sobolev inequality.”
The author of the article introduced in [Calc. Var. Partial Differ. Equ. 48, No. 3–4, 507–526 (2013; Zbl 1275.49081)] a “one-parameter family of geometric invariants which interpolate between the Yamabe constant and the \(\nu\)-entropy”. Those invariants are called weighted Yamabe constants.
“The purpose of the article is to study to what extend these invariants interpolate between the Yamabe constant and the \(\nu\)-entropy, focusing on issues related to the problem of finding minimizers of the weighted Yamabe quotients.”
The following theorems are shown:
“Let \((M^n,g,e^{-\phi} \mathrm{dvol},m)\) be a compact smooth metric measure space. Then \[ \Lambda[g,e^{-\phi}\mathrm{dvol},m] \leq \Lambda[\mathbb{R}^n,\mathrm{d}x^2,\mathrm{dvol},m] = \Lambda_{m,n}. \] Moreover, if the above inequality is strict, then there exists a positive function \(w \in C^\infty(M)\) such that \[ \mathcal{Q}(w) = \Lambda[g,e^{-\phi}\mathrm{dvol},m]." \] Here, \(\mathcal{Q}\) is the weighted Yamabe quotient, \(\Lambda[g,e^{-\phi}\mathrm{dvol},m]\) denotes the weighted Yamabe constant and \(\Lambda_{m,n}\) is a constant depending on \(m\) and \(n\).
Let \((M^n,g,e^{-\phi}\mathrm{dvol},m)\) be a compact smooth metric measure space such that \(m \in \mathbb{N} \cup \{0,\infty\}\). If \[ \Lambda[g,e^{-\phi}\mathrm{dvol},m] = \Lambda[\mathbb{R}^n,\mathrm{d}x^2,\mathrm{dvol},m], \] then \(m \in \{0,1\}\) and \((M^n,g,e^{-\phi}\mathrm{dvol},m)\) is conformally equivalent to \((S^n,g_0,\mathrm{dvol},m)\) for a metric \(g_0\) of constant sectional curvature. In particular, there exists a positive function \(w \in C^\infty(M)\) such that \[ \mathcal{Q}(w) = \Lambda[g,e^{-\phi}\mathrm{dvol},m]. \]
It is shown that “There does not exist a minimizer for the weighted Yamabe constant of \((S^n,g_0,1^{1/2}\mathrm{dvol})\)”.
Reviewer: Loreno Heer (Zürich)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53A30 | Conformal differential geometry (MSC2010) |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
