Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds (Q6611124)

The main objects of this paper are Riemannian manifolds \(M\) of dimension \(n\), Ricci curvature \(\geq K\), and diameter \(\leq D\). For non-negative \(\delta\) and positive \(C, \tau\), the authors define \((\delta ; C, \tau )\)-harmonic maps: they are harmonic maps \(\Phi : M \to \mathbb{R}^n / \mathbb{Z}^n\) with energy \(\leq C\), non-degeneracy controlled by \(\tau\), and \(\delta\)-close to being affine. This is chosen in such a way that if a manifold \(M\) admits a \(( 0 ; C, \tau )\)-harmonic map \(\Phi : M \to \mathbb{R}^n / \mathbb{Z}^n\), then \(M\) is a flat torus and \(\Phi\) is an affine covering map. The paper studies the stability of this characterization.\N\NThroughout this paper, \(\Psi (\delta ; A, B, C)\) denotes a non-negative quantity depending on \(\delta, A, B, C\), that goes to \(0\) as \(\delta\) goes to \(0\) provided \(A\), \(B\), \(C\) are fixed.\N\NTheorem 1.4 is the main result of this paper. It (almost!) establishes that \(M\) is Gromov-Hausdorff close (and consequently diffeomorphic) to a flat \(n\)-torus if and only if it admits \((\delta ; C, \tau)\)-harmonic maps with \(\delta\) small. More concretely: assume \(M\) is an \(n\)-dimensional Riemannian manifold of Ricci curvature \(\geq K\) and diameter \(\leq D\).\N\begin{itemize}\N\item If \(M\) admits a \((\delta ; C, \tau)\)-harmonic map \(\Phi\), then it is \(\Psi \)-Gromov-Hausdorff-close to a flat torus of volume \(\geq c \) for some \(\Psi (\delta; C, \tau , K, n, D)\), \(c(C, \tau , K, n, D)\).\N\item If \(M\) is \(\delta\)-GH-close to a flat torus of volume \(\geq \tau\), and has almost-non-negative Ricci in an integral sense, then it admits a \((\Psi ; C_1, C_2)\)-harmonic diffeomorphism \(\Phi : M \to \mathbb{R}^n / \mathbb{Z}^n\) for some \(\Psi (\delta ; \tau , K, n, D)\) and some \(C_1\), \(C_2\) depending on \(\tau , K, n, D \).\N\end{itemize}\N\NIn the second part, it is conjectured that the almost-non-negative-Ricci condition on \(M\) is unnecessary. The proof provides multiple technical contributions of their own interest.\N\NTheorem 3.1 establishes that if a non-collapsing sequence of \(n\)-dimensional Riemannian manifolds of Ricci curvature \(\geq K\) and diameter \(\leq D\) admits an \(L^2\)-bounded sequence of harmonic vector fields, then one can find a subsequence for which the vector fields converge strongly to a vector field in a limit space. This is in the spirit of previous work of the first author ([\textit{S. Honda}, Elliptic PDEs on compact Ricci limit spaces and applications. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1400.53031)]).\N\NTheorem 4.1 shows that if an RCD\((K,N)\) space \(M\) of dimension \(n\) admits \(n\) \(L^2\)-orthonormal vector fields in \(H^{1,2}_C(TM)\) with vanishing covariant derivative and divergence, then \(M\) is a flat torus. This generalizes a result of \textit{N. Gigli} and \textit{C. Rigoni} [Calc. Var. Partial Differ. Equ. 57, No. 4, Paper No. 104, 39 p. (2018; Zbl 1404.53057)].\N\NThe proof uses results and ideas from [\textit{B.-X. Han}, Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 113, 35 p. (2018; Zbl 1401.30068)] to show that the regular Lagrangian flows associated to the given vector fields have representatives that are measure-preserving isometries. After this, it follows from [\textit{N. Gigli} and \textit{C. Rigoni}, loc. cit.] that these flows provide a transitive \(\mathbb{R}^n\)-action by measure-preserving isometries, finishing the proof.\N\NTheorem 5.1 is a stability version of Theorem 4.1. If an \(n\)-dimensional Riemannian manifold \(M\) of Ricci curvature \(\geq K\), diameter \(\leq D\), and volume \(\geq \tau\) admits \(n\) \(L^2\)-orthonormal harmonic vector fields with \(L^2\)-small covariant derivative, then \(M\) is Gromov-Hausdorff close to a flat torus.\N\NThe proof is done by contradiction. Given a contradicting sequence, Theorem 3.1 allows one to find suitable limit vector fields. Using the structural properties of the limit space obtained in [\textit{J. Cheeger} et al., Ann. Math. (2) 193, No. 2, 407--538 (2021; Zbl 1469.53083)], it is established that such limit vector fields have the necessary regularity to apply Theorem 4.1 and conclude.\N\NTheorem 5.3 is another torus-characterization stability result. It (almost!) shows that \(M\) is GH-close to a flat torus if and only if one can \(L^p\)-approximate the Riemannian metric by \(\sum_{j=1}^n V_j \otimes V_j\) with \(V_j\) harmonic vector fields. Just like Theorem 1.4, the ``if'' part requires \(M\) to have almost-non-negative Ricci in an integral sense.\N\NThe proof of the ``if'' part relies on Theorem 5.1, while the ``only if'' part uses the spectral convergence studied in [\textit{S. Honda}, J. Funct. Anal. 273, No. 5, 1577--1662 (2017; Zbl 1368.53027)].\N\NThe proof of the first part of Theorem 1.4 is similar to the one of Theorem 5.1, and uses results of Honda and Peng to give a detailed description of \(\Phi\). The maps for the second part are obtained by applying the Eells-Sampson heat flow to the canonical diffeomorphisms obtained in [\textit{S. Honda} and \textit{Y. Peng}, Manuscr. Math. 172, No. 3--4, 971--1007 (2023; Zbl 1525.53054)]. Then Theorem 5.3 is used to show that they are indeed \((\Psi , C_1, C_2)\)-harmonic.\N\NThe conditions on the first part of Theorem 1.4 are satisfied in a couple of notable situations: when \(M\) has almost-non-negative Ricci curvature everywhere, the \((\delta ; C, \tau )\)-condition can be relaxed (Theorem 1.8), and when \(M\) is homeomorphic to \(\mathbb{R}^3 / \mathbb{Z}^3\) and has almost-non-negative scalar curvature in an integral sense, it admits \((\delta ; C, \tau )\)-harmonic maps (Theorem 1.7). It is conjectured that Theorem 1.7 holds in all dimensions. Its proof uses the results of Simon--Topping on Ricci flow [\textit{M. Simon} and \textit{P. M. Topping}, Geom. Topol. 25, No. 2, 913--948 (2021; Zbl 1470.53083)], together with \textit{D. L. Stern}'s inequality [J. Differ. Geom. 122, No. 2, 259--269 (2022; Zbl 1514.53110)].