Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.(English)Zbl 07811955
The authors study qualitative stability results for Sobolev inequalities on (smooth and non-smooth) spaces verifying lower Ricci curvature bounds.
The first main result states the following. Let \((M,g)\) be a closed \(n\)-dimensional Riemannian manifold (with \(n>2\)) satisfying \(\mathrm{Ric}_g\geq(n-1)g\), where \(\mathrm{Ric}_g\) denotes the Ricci tensor of \((M,g)\). Then the following Sobolev inequality holds:\[\|u\|_{L^{2^*}}^2\leq\frac{2^*-2}{n}\|\nabla u\|_{L^2}^2+\|u\|_{L^2}^2\quad\text{ for every }u\in W^{1,2}(M),\]where \(2^*:=\frac{2n}{n-2}\) denotes the Sobolev conjugate of \(2\) and the norms are computed with respect to the renormalized volume measure; see [S. Ilias, Ann. Inst. Fourier 33, No. 2, 151–165 (1983;Zbl 0528.53040)]. The authors prove that ‘functions that almost satisfy the equality in the Sobolev inequality are close to a spherical bubble’, i.e., to a function of the form\[w_{a,b,z}(\cdot):=\frac{a}{(1-b\cos(\mathsf{d}_g(\cdot,z)))^{\frac{n-2}{2}}} \quad\text{ for }a\in\mathbb R,\,b\in[0,1),\,z\in M,\]where \(\mathsf{d}_g\) denotes the distance induced by the metric \(g\). More precisely, Theorem 1.1 states that if \(\varepsilon>0\) is given and there exists a non-constant function \(u\in W^{1,2}(M)\) satisfying\[\frac{\|u\|_{L^{2^*}}^2-\|u\|_{L^2}^2}{\|\nabla u\|_{L^2}^2}>\frac{2^*-2}{n}-\delta\]for a suitable constant \(\delta>0\) that depends only on \(\varepsilon\) and \(n\), then there exist \(a\in\mathbb R\), \(b\in[0,1)\) and \(z\in M\) such that\[\frac{\|\nabla(u-w_{a,b,z})\|_{L^2}+\|u-w_{a,b,z}\|_{L^{2^*}}}{\|u\|_{L^2}}\leq\varepsilon.\]As an application of Theorem 1.1, the authors obtain in Corollary 1.3 a stability result for minimizing Yamabe metrics.
The second main result, Theorem 1.4, is a qualitative stability result for (non-compact) Riemannian manifolds of dimension \(n>2\) having non-negative Ricci curvature and Euclidean volume growth, the latter meaning that the asymptotic volume ratio\[\mathrm{AVR}(M):=\lim_{R\to\infty}\frac{\mathrm{Vol}_g(B(x,r))}{\omega_n R^n}\](which is independent of the chosen point \(x\in M\)) is strictly positive. The authors show, roughly speaking, that every non-zero function \(u\in\dot W^{1,2}(M)\) that almost satisfies the sharp Euclidean-type Sobolev inequality (see [Z. M. Balogh andA. Kristály, Math. Ann. 385, No. 3–4, 1747–1773 (2023;Zbl 1514.53079)]) is ‘close’ to a Euclidean bubble.
The above stability results are actually obtained in the more general setting of \(\mathsf{RCD}\) spaces, i.e., of metric measure spaces verifying synthetic lower Riemannian Ricci curvature bounds, see Theorems 8.1 and 8.4. In fact, possibly singular spaces have an essential role in the proof strategy of the main results. Indeed, the proof relies on a reductio ad absurdum argument, whose main ingredients are a generalized Lions’ concentration-compactness principle on varying spaces (where singular and non-compact limits may appear) and rigidity results for Sobolev inequalities on singular spaces.
The first main result states the following. Let \((M,g)\) be a closed \(n\)-dimensional Riemannian manifold (with \(n>2\)) satisfying \(\mathrm{Ric}_g\geq(n-1)g\), where \(\mathrm{Ric}_g\) denotes the Ricci tensor of \((M,g)\). Then the following Sobolev inequality holds:\[\|u\|_{L^{2^*}}^2\leq\frac{2^*-2}{n}\|\nabla u\|_{L^2}^2+\|u\|_{L^2}^2\quad\text{ for every }u\in W^{1,2}(M),\]where \(2^*:=\frac{2n}{n-2}\) denotes the Sobolev conjugate of \(2\) and the norms are computed with respect to the renormalized volume measure; see [S. Ilias, Ann. Inst. Fourier 33, No. 2, 151–165 (1983;Zbl 0528.53040)]. The authors prove that ‘functions that almost satisfy the equality in the Sobolev inequality are close to a spherical bubble’, i.e., to a function of the form\[w_{a,b,z}(\cdot):=\frac{a}{(1-b\cos(\mathsf{d}_g(\cdot,z)))^{\frac{n-2}{2}}} \quad\text{ for }a\in\mathbb R,\,b\in[0,1),\,z\in M,\]where \(\mathsf{d}_g\) denotes the distance induced by the metric \(g\). More precisely, Theorem 1.1 states that if \(\varepsilon>0\) is given and there exists a non-constant function \(u\in W^{1,2}(M)\) satisfying\[\frac{\|u\|_{L^{2^*}}^2-\|u\|_{L^2}^2}{\|\nabla u\|_{L^2}^2}>\frac{2^*-2}{n}-\delta\]for a suitable constant \(\delta>0\) that depends only on \(\varepsilon\) and \(n\), then there exist \(a\in\mathbb R\), \(b\in[0,1)\) and \(z\in M\) such that\[\frac{\|\nabla(u-w_{a,b,z})\|_{L^2}+\|u-w_{a,b,z}\|_{L^{2^*}}}{\|u\|_{L^2}}\leq\varepsilon.\]As an application of Theorem 1.1, the authors obtain in Corollary 1.3 a stability result for minimizing Yamabe metrics.
The second main result, Theorem 1.4, is a qualitative stability result for (non-compact) Riemannian manifolds of dimension \(n>2\) having non-negative Ricci curvature and Euclidean volume growth, the latter meaning that the asymptotic volume ratio\[\mathrm{AVR}(M):=\lim_{R\to\infty}\frac{\mathrm{Vol}_g(B(x,r))}{\omega_n R^n}\](which is independent of the chosen point \(x\in M\)) is strictly positive. The authors show, roughly speaking, that every non-zero function \(u\in\dot W^{1,2}(M)\) that almost satisfies the sharp Euclidean-type Sobolev inequality (see [Z. M. Balogh andA. Kristály, Math. Ann. 385, No. 3–4, 1747–1773 (2023;Zbl 1514.53079)]) is ‘close’ to a Euclidean bubble.
The above stability results are actually obtained in the more general setting of \(\mathsf{RCD}\) spaces, i.e., of metric measure spaces verifying synthetic lower Riemannian Ricci curvature bounds, see Theorems 8.1 and 8.4. In fact, possibly singular spaces have an essential role in the proof strategy of the main results. Indeed, the proof relies on a reductio ad absurdum argument, whose main ingredients are a generalized Lions’ concentration-compactness principle on varying spaces (where singular and non-compact limits may appear) and rigidity results for Sobolev inequalities on singular spaces.
Reviewer: Enrico Pasqualetto (Jyväskylä)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Cite
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