arXiv subjects

Junqi Lai

Publications and source records attributed to Junqi Lai.

6 recordsLinked to original sources

arXiv2026

Singular Rotational Self-Similar Tori for Odd $\sigma_k$-Curvature Flows

For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has H\"older regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\langle X,\nu\rangle=-\sigma_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

math.DG↗

arXiv2026

Examples of compact embedded mean convex $\lambda$-hypersurfaces

There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $\lambda$-hypersurfaces, if $\lambda<0$, we constructed compact convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $\lambda>0$, we construct a compact mean convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $\lambda>0$, there are no compact convex embedded $\lambda$-hypersurfaces which are diffeomorphic to spheres except a round sphere.

math.DG↗

arXiv2026

Minimal hypersurfaces in spheres generated by isoparametric foliations

We investigate the existence of minimal hypersurfaces in $\mathbb{S}^{n+1}$ that are generated by the isoparametric foliation of a subsphere $\mathbb{S}^n$. By considering a generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, we reduce the minimal surface equation to an ordinary differential equation. We prove that this construction yields a closed embedded minimal hypersurface for any choice of isoparametric hypersurface $M \subset \mathbb{S}^n$. The resulting hypersurfaces have the topological type $S^1 \times M$, extending the known examples of minimal hypertori ($S^1\times S^k\times S^k$ and $S^1\times S^k\times S^l$) to a broader class of topologies determined by isoparametric structures.

math.DG↗

arXiv2025

Embedded constant mean curvature hypertori in the $2n$-sphere

Brendle proved Lawson conjecture about minimal embedded torus in the round three-dimensional sphere. Carlotto and Schulz constructed a minimal embedded three-dimensional hypertorus in the round four-dimensional sphere and conjectured that their hypertorus is a unique minimal embedded three-dimensional hypertorus in the round four-dimensional sphere. In this paper, we construct two different constant mean curvature embedded $(2n-1)$-dimensional hypertori (that is, topological type $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$) which have the same negative mean curvature $H$ in the round $2n$-dimensional sphere $\mathbb{S}^{2n}(1)$ .

math.DG↗

arXiv2024

Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces

In the paper, we construct, for $λ>0$, complete embedded and non-convex $λ$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $λ$-hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle \cite{B} affirmatively. Furthermore, for a fixed $λ<0$ which may have small $|λ|$, we can construct two compact embedded $λ$-hypersurfaces which are diffeomorphic to $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$, but they are not isometric to each other.

math.DG↗

arXiv2023

Examples of compact embedded $λ$-hypersurfaces

In the paper, we construct compact embedded $λ$-hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. Hence, one can not expect to have Alexandrov type theorem for $λ$-hypersurfaces.

math.DG↗