Rigidity of the round cylinders in Ricci shrinkers (Q6562501)

The paper under review concerns the Ricci flow theory and continues a recent series of the authors' research papers devoted to studying the so-called Ricci shrinkers.\N\NBy definition, a Ricci shrinker is a triple \((M^n, g, f)\), where \(M^n\) is a smooth manifold, \(g\) is a Riemannian metric, and \(f\) is a smooth function satisfying \[ \mathrm{Ric} + \mathrm{Hess} f = \frac{1}{2} g,\] where the potential function \(f\) is normalized so that \(R+\vert\nabla f\vert^2 = f\)\N\NIn dimensions 2 and 3, the only Ricci shrinkers are \(\mathbb R^2\), \(\mathbb S^2\), \(\mathbb R^3\), \(\mathbb S^3\), \(\mathbb S^2 \times\mathbb R^1\) and their quotients.\N\NEvidently, in dimension \(n>3\) the classification of Ricci shrinkers is still a quite challenging open problem.\N\NTo approach this problem, the authors consider the moduli space of Ricci shrinkers, \(\mathcal M\), equipped with the pointed-Gromov-Hausdorff distance, and discuss whether a given Ricci shrinker is rigid in the sense that there exists no nearby Ricci shrinker other than itself. In the compact case examples of such rigidity are provided by \(\mathbb S^n/\Gamma\), \(\mathbb CP^{2n}\), \(\mathbb S^2\times \mathbb S^2\). In the non-compact case the problem seems to be more difficult, the only non-compact Ricci shrinkers known to be rigid were the Gaussian solitons \((\mathbb R^n; g_E)\).\N\NPaving the way to a complete description (classification) of Ricci shrinkers in large dimensions, the authors explore the rigidity of the round cylinder \((\mathbb S^{n-1}\times\mathbb R, g_c)\).\N\NNamely, given a Ricci shrinker \((M^n, g, f)\), let \(p\) be the point where \(f\) achieves its minimum value. Then one can consider the pointed-Gromov-Hausdorff distance \(d_{pGH}\) from \((M^n, g, p)\) to \((\mathbb S^{n-1}\times\mathbb R, g_c, p_0)\), where \(p_0\) is a fixed point of \(\mathbb S^{n-1}\times\mathbb R\). It turns out that there exists a constant \(\varepsilon (n) > 0\) such that if the distance \(d_{pGH}\) between \((M^n, g, p)\) and \((\mathbb S^{n-1}\times\mathbb R, g_c, p_0)\) is less than \(\varepsilon\), then \((M^n, g)\) is isometric to \((\mathbb S^{n-1}\times\mathbb R, g_c)\). Thus, the round cylinders \((\mathbb S^{n-1}\times\mathbb R, g_c)\) are proved to be rigid in the moduli space of Ricci shrinkers.\N\NIt is conjectured that similar rigidity phenomena hold true for quotients \((\mathbb (S^{n-1} / \Gamma ) \times\mathbb R, g_c)\) and for round cylinders \((\mathbb S^{n-k}\times\mathbb R^k, g_c)\) with \(2\leq k\leq n-2\) as well.