• Ximin Liu

      Articles written in Proceedings – Mathematical Sciences

    • Hypersurfaces Satisfying $L_rx = Rx$ in Sphere $\mathbb{S}^{n+1}$ or Hyperbolic Space $\mathbb{H}^{n+1}$

      Biaogui Yang Ximin Liu

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      In this paper, using the method of moving frames, we consider hypersurfaces in Euclidean sphere $\mathbb{S}^{n+1}$ or hyperbolic space $\mathbb{H}^{n+1}$ whose position vector 𝑥 satisfies $L_r x=Rx$, where $L_r$ is the linearized operator of the $(r+1)$-th mean curvature of the hypersurfaces for a fixed $r=0,\ldots,n-1,R\in \mathbb{R}^{(n+2)\times(n+2)}$. If the 𝑟-th mean curvature $H_r$ is constant, we prove that the only hypersurfaces satisfying that condition are 𝑟-minimal $(H_{r+1}\equiv 0)$ or isoparametric. In particular, we locally classify such hypersurfaces which are not 𝑟-minimal.

    • Nonsmoothable Involutions on Spin 4-Manifolds

      Changtao Xue Ximin Liu

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      Let 𝑋 be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to $n(-E_8)\oplus mH$, where 𝐻 is the hyperbolic form. In this paper, we prove that for 𝑛 such that $n\equiv 2\mathrm{mod} 4$, there exists a locally linear pseudofree $\mathbb{Z}_2$-action on 𝑋 which is nonsmoothable with respect to any possible smooth structure on 𝑋.

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