arXiv subjects
Publications and source records attributed to Zhizhang Xie.
We prove Llarull's scalar curvature rigidity theorem for spheres and scalar-mean rigidity for strictly convex Euclidean domains in all dimensions without the spin assumption.
We prove Schoen's conjecture on $L^\infty$-metrics for tori. More precisely, we show that any $L^\infty$-metric on a torus that is smooth and has non-negative scalar curvature away from an embedded submanifold with codimension at least three extends to a smooth flat metric. We also prove a LLarull-type theorem for $L^\infty$-metric. Our proof uses weighted scalar curvature and the relative index theorem.
Consider a sequence of metrics $g_i$ on the torus whose members have uniform lower bounds on their first stable systoles and Ricci curvatures, and have a uniform upper bound on their diameters. If the $L^1$ norm of the negative part of the scalar curvatures vanishes along this sequence, then we show there is a subsequence of the metrics which converges in the measured-Gromov-Hausdorff topology to a flat metric on the torus. Something analogous holds for sequences of asymptotically flat spin Riemannian manifolds with a uniform lower bound on Ricci curvature and nonnegative scalar curvature: if the ADM masses of the distinguished ends tend to zero along the sequence, then the manifolds converge to Euclidean space in the pointed measured Gromov-Hausdorff sense.
In this paper, we prove Gromov's simplicial volume vanishing conjecture for closed manifolds with spin universal cover. More precisely, we show that if a closed oriented manifold admits a metric of nonnegative scalar curvature and its universal cover is spin, then its simplicial volume vanishes. In particular, a closed oriented aspherical manifold with nonzero simplicial volume admits no metric of nonnegative scalar curvature.
We prove a sharp $2$-systolic inequality for four-dimensional products $\mathbb S^2\times P$, where $P\subset\mathbb R^2$ is an arbitrary convex polygon. Let $h=g_{\mathbb S^2}+g_{\mathrm{eu}}$ be the standard product metric. If a Riemannian metric $g$ on $\mathbb S^2\times P$ has scalar curvature $\geq σ>0$, nonnegative mean curvature on every codimension one face, and dihedral angles no larger than the corresponding dihedral angles of $h$, then both its homotopy and homology $2$-systoles are at most $ \frac{8π}σ.$ This confirms a conjecture of Gromov.
In this article, we present a self-contained proof of Gromov's dihedral rigidity conjecture on scalar curvature in the three-dimensional case. The proof avoids many of the technical complications that arise in higher dimensions, while still illustrating the essential ideas of the general approach developed in arXiv:2112.01510 (version 6) and arXiv:2203.09511. It is significantly shorter than the proof of the general case and is intended to be more accessible.
We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslashΣ$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $Σ$ is a closed subset of $\mathbb S^n$ of codimension at least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of wrapping property was introduced by the second author for studying related scalar curvature rigidity problems on spheres. For example, any closed subset of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb S^n$ satisfy the wrapping property. The same techniques also apply to prove an analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth away from certain subsets of codimension at least $\frac{n}{2}+1$. As a corollary, we obtain a positive mass theorem for complete asymptotically flat spin manifolds with arbitrary ends for $L^\infty$ metrics.
The main scalar-mean extremality and rigidity results in the existing literature concern manifolds whose curvature operators are nonnegative, or warped product spaces with a log-concave warping function whose leaves carry metrics of nonnegative curvature operator. In this paper, we establish scalar-mean extremality and rigidity theorems for a broad class of Riemannian manifolds with boundary whose metrics are conformal to ones with nonnegative curvature operator. In particular, our results extend these theorems beyond the warped product setting and yields new families of manifolds exhibiting scalar-mean extremality and rigidity.
In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave.
In this article, we prove a strong relative Novikov conjecture for any pair of groups that are coarsely embeddable into Hilbert space.
We prove an $\ell^p$-version of the coarse Baum-Connes conjecture for spaces that coarsely embedds into $\ell^q$-spaces for any $p$ and $q$ in $[1,\infty)$.
In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator $K$-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.
For each orientation-preserving homotopy equivalence between two closed oriented smooth manifolds, there are mainly two different approaches to the higher $ρ$ invariant associated to this homotopy equivalence. In this article, we show that these two definitions of the higher $ρ$ invariant are equivalent.
We investigate the influence of uniformly positive scalar curvature on the size of a non-collapsed Ricci limit space coming from a sequence of $n$-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. We prove that such a limit space splits at most $n-2$ lines or $\mathbb{R}$-factors. When this maximal splitting occurs, we obtain a uniform upper bound on the diameter of the non-splitting factor. Moreover, we obtain a volume gap estimate and a volume growth order estimate of geodesic balls on such manifolds.
Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $μ$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.
We prove a quantitative upper bound on the filling radius of complete, spin manifolds with uniformly positive scalar curvature using the quantitative operator $K$-theory and index theory.
In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with nonnegative curvature operators and nonvanishing Euler characteristics, flat tori, round spheres and their direct products. In particular, we obtain the scalar curvature extremality and rigidity for certain degenerate toric bands and also for round spheres with two antipodal points removed. This answers positively the corresponding questions of Gromov in all dimensions.
Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method.