Spectral constant rigidity of warped product metrics (Q6586634)

Llarull's rigidity theorem in Riemannian geometry [\textit{M. Llarull}, Math. Ann. 310, No. 1, 55--71 (1998; Zbl 0895.53037)] implies the following. Let \(g\) be any Riemannian metric on the \(n\)-dimensional sphere \(\mathbb{S}^n\) and denote by \(R_g\) its scalar curvature; if \(g\geq \overline{g}\), where \(\overline{g}\) is the standard round metric of sectional curvature \(1\), and \(R_g\geq R_{\overline{g}}=n(n-1)\), then \((\mathbb{S}^n,g)\) is isometric to \((\mathbb{S}^n,\overline{g})\).\N\NThe authors generalize Llarull's result as follows: the manifold \(\mathbb{S}^n\) is replaced by any open manifold (noncompact manifold without boundary) \(M\) and the scalar curvature bound is replaced by a lower bound on some spectral constant \(\Lambda_c\), which is defined by\N\[\N\Lambda_c:=\inf\left\{ \int_M(|\nabla u|^2 + cR_gu^2)\mathrm{dv}_g,\;u\in H_0^1(M),\,\int_M u^2\mathrm{dv}_g=1\right\},\N\]\Nwhere \(R_g\) denotes the scalar curvature of \((M,g)\). The model manifold to which \(M\) is compared is the warped product \(\left((0,\frac{\pi}{b})\times\mathbb{S}^{n-1},g_0:=d\theta\otimes d\theta+a^2\sin(b\theta)^2g_{\mathbb{S}^{n-1}}\right)\), where \(a\) and \(b\) are two positive real parameters defined in terms of \(\Lambda,n\) and \(c\), see the definition on page 2. Here \((0,\frac{\pi}{b})\times\mathbb{S}^{n-1}\) is considered as \(\mathbb{S}^n\) minus \(2\) points in spherical polar coordinates.\N\NThe main theorem of the paper, which is called the Llarull spectral rigidity theorem, reads as follows (Theorem 1.3): Let \(M\) be an \(n\)-dimensional, noncompact, connected spin manifold without boundary. Fix two positive constants \(c>\frac{1}{4}\) and \(\Lambda\), and let \(g\) be a Riemannian metric on M such that \(\Lambda_c(g) \geq \Lambda_c (g_0 ) = \Lambda\). Assume that \(\Phi \colon (M, g)\longrightarrow\left((0, \frac{\pi}{b})\times\mathbb{S}^{n-1},g_0\right)\) is a smooth map which is proper, has nonzero degree and is \(1\)-Lipschitz. Then \(\Phi\) is a Riemannian isometry and \(\Lambda_c = \Lambda\).\N\NAs a consequence, in the case where \(a=b=1\), a recent statement by \textit{C. Bär} et al. [SIGMA, Symmetry Integrability Geom. Methods Appl. 20, Paper 035, 26 p. (2024; Zbl 07846465)] can be recovered (Theorem 1.4): if \(n\geq3\), the map \(\Phi \colon (M, g)\longrightarrow\left((0, \frac{\pi}{b})\times\mathbb{S}^{n-1},g_0\right)\) satisfies the above assumptions and the spectral condition is replaced by \(R_g\geq n(n-1)\), then \(\Phi\) must be an isometry.\N\NThe main result can be extended to the case where the infimum takes a supplementary penalizing term into account (Theorem 1.5), that is, given a real-valued function \(\mu\) on \(M\), the infimum of \(\displaystyle\int_M(|\nabla u|^2 + cR_gu^2-\mu u^2)\mathrm{dv}_g\) over the same function space is considered.\N\NTheorem 1.3 as well as the above-mentioned extension (Theorem 1.5) are proved by spinorial methods in all dimensions \(n\geq3\), see Section 2. In dimension \(3\), an alternative proof based on spacetime harmonic functions is given (Section 3). These detailed calculations that are not carried out in the proofs can be found in an appendix.