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Xiaoxiang Chai

Publications and source records attributed to Xiaoxiang Chai.

At least 19 recordsLinked to original sources
arXiv2026

Rigidity and weak notions of spectral scalar curvature

We give a weak formulation of spectral scalar curvature bounded from below via establishing a dihedral rigidity result. Given some special convex polyhedron, if another metric are of non-negative spectral scalar curvature in the interior, with weighted mean-convex faces, and with its dihedral angles less than or equal to their flat polyhedral model everywhere along the edges, then the metric must be flat. This is motivated by Gromov's definition of a weak notion of non-negative scalar curvature.

math.DG↗
arXiv2026

Band width estimates with lower spectral curvature bounds

In this work, we use the warped \( μ\)-bubble method to study the consequences of a spectral curvature bound. In particular, with a lower spectral Ricci curvature bound and a lower spectral scalar curvature bound, we show that the band width of a torical band is bounded above. We also obtain some rigidity results.

math.DG↗
arXiv2026

Some rigidity theorems for spectral curvature bounds

We investigate the geometric implications of spectral curvature bounds, extending classical rigidity results in scalar curvature geometry to the spectral setting. By systematically employing the warped $μ$-bubble method, we show classification theorems for stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature, and we establish band width estimates for both spectral Ricci and spectral scalar curvatures. Furthermore, we prove some splitting theorems under spectral curvature conditions, including a spectral version of the Geroch conjecture for manifolds with arbitrary ends and a result related to the Milnor conjecture.

math.DG↗
arXiv2026

Llarull type theorems for bands in Three and Four dimensions

Llarull's theorem asserts that the scalar curvature and the metric on the $n$-sphere cannot be bounded below at the same time by those of the standard $n$-sphere. Using the warped $μ$-bubble method, we develop Llarull type theorems for three and four-dimensional bands with spectral scalar curvature bounds.

math.DG↗
arXiv2025

Scalar curvature rigidity of domains in a 3-dimensional warped product

A warped product with a spherical factor and a logarithmically concave warping function satisfies a scalar curvature rigidity of the Llarull type. We develop a scalar curvature rigidity of the Llarull type for a general class of domains in a three dimensional spherical warped product. In the presence of rotational symmetry, we identify this class of domains as those satisfying a boundary condition analogous to the logarithmic concavity of the warping function.

math.DG↗
arXiv2024

Scalar curvature rigidity of parabolic convex polytopes in hyperbolic space

In odd dimensions, we prove a scalar curvature rigidity for parabolic convex polytopes in hyperbolic space enclosed by linear planes in the Poincare upper half-space model and convex with respect to the conformally related flat metric. Our method is based on spinor techniques and relies on the recent smoothing constructions of Brendle-Wang. We also prove a Llarull type rigidity for bounded smooth parabolic convex domains and a dihedral rigidity for polytopal initial data sets with dominant energy conditions.

math.DG↗
arXiv2024

Scalar curvature comparison of rotationally symmetric sets

Let $(M, g)$ be a compact 3-manifold with nonnegative scalar curvature $R_g\geq 0$. The boundary $\partial M$ is diffeomorphic to the boundary of a rotationally symmetric and weakly convex body $\bar{M}$ in $\mathbb{R}^3$. We call $(\bar{M}, δ)$ a model or a reference. Let $H_{\partial M}$ and $\bar{H}_{\partial M}$ be respectively the mean curvatures of $\partial M$ in $(M, g)$ and $\partial M$ in $(\bar{M}, δ)$, $σ$ and $\barσ$ be the induced metric from $g$ and $δ$. We show that for some classes of $\partial M$, if $H_{\partial M} \geq \bar{H}_{\partial M}$, $σ\geq \barσ$ and the dihedral angles at the nonsmooth part of $\partial M$ are no greater than the model, then $M$ is flat. We also generalize this result to the hyperbolic case and some spaces with $\mathbb{S}^1$-symmetry. Our approach is inspired by Gromov.

math.DG↗
arXiv2024

Initial data set rigidity results for polyhedra

Using spinors, we show a dihedral type rigidity for polyhedral initial data sets. This rigidity connects spacetime positive mass theorem, dihedral rigidity and capillary marginally trapped surfaces. Our method is to extend the rigidity analysis of spacetime positive mass theorem due to Beig-Chrusciel to the settings of a twisted spinor bundle.

math.DG↗
arXiv2024

Spectral constant rigidity of warped product metrics

A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere $\mathbb{S}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function.

math.DG↗
arXiv2024

Scalar curvature rigidity of domains in a warped product

By exploiting the conformality of a warped product metric with a direct product metric, we develop a new connection on a twisted spinor bundle and its associated Dirac operator. We obtain a Llarull type scalar curvature rigidity for a general class of domains in a warped product. Also, we are able to address Gromov dihedral rigidity in hyperbolic space assuming matching angles.

math.DG↗
arXiv2024

A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

We prove a new Minkowski type formula for capillary hypersurfaces supported on totally geodesic hyperplanes in hyperbolic space. It leads to a volume-preserving flow starting from a star-shaped initial hypersurface. We prove the long-time existence of the flow and its uniform convergence to a $θ$-totally umbilical cap. Additionally, we establish that a $θ$-totally umbilical cap is an energy minimizer for a given enclosed volume.

math.DG↗
arXiv2023

A tilted spacetime positive mass theorem

We show a spacetime positive mass theorem for asymptotically flat initial data sets with a noncompact boundary. We develop a mass type invariant and a boundary dominant energy condition. Our proof is based on spinors.

math.DG↗
arXiv2023

A curvature estimate for stable marginally outer trapped hypersurface with a free boundary

A marginally outer trapped hypersurface is a generalization of minimal hypersurfaces originated from general relativity. We show a curvature estimate for stable marginally outer trapped hypersurfaces up to the free boundary satisfying a uniform area bound. Our proof is based on an iteration argument. The curvature estimate was previously known via a blowup argument for stable minimal hypersurfaces.

math.DG↗
arXiv2022

Dihedral rigidity in hyperbolic 3-space

We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.

math.DG↗
arXiv2022

The mass of an asymptotically hyperbolic end and distance estimates

Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with compact boundary, we prove the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a function defined using distance estimates. As an application, the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by $-(n-1)$ or the scalar curvature satisfies $R_g\geq (-1+κ)n(n-1)$ for any positive constant $κ$ less than one.

math.DG↗
arXiv2022

Band width estimates of CMC initial data sets

We generalize a band width estimate of Gromov to CMC initial data sets. We give three independent proofs: via the stability of a hypersurface with prescribed null expansion, via a perturbation of the spacetime harmonic function and via the Dirac operator.

math.DG↗
arXiv2022

Inverse mean curvature flow with a free boundary in hyperbolic space

We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.

math.DG↗