Kai Xu

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20242026

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math.DG2025

Connected sum of manifolds with spectral Ricci lower bounds

Gioacchino Antonelli, Kai Xu

Let $n > 2$ , $γ > \frac{n - 1}{n - 2}$ , and $λ \in R$ . We prove that if $M$ and $N$ are two smooth $n$ -manifolds that admit a complete Riemannian metric satisfying \[-γΔ+ \mathr…

math.DG2024

A sharp spectral splitting theorem

Gioacchino Antonelli, Marco Pozzetta, Kai Xu

We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold $M$ of dimension $n \geq 2$ has at lea…

math.DG2024

Inverse mean curvature flow with outer obstacle

Kai Xu

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of…

math.DG2024

New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry

Gioacchino Antonelli, Kai Xu

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M, g)$ of dimension $n \geq 3$ satisfies…