arXiv subjects

Jiuzhou Huang

Publications and source records attributed to Jiuzhou Huang.

5 recordsLinked to original sources

arXiv2025

Self-similar solutions of semilinear heat equations with positive speed

We classify the smooth self-similar solutions of the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a positive constant after rescaling at the blow-up point for all $p>1$.

math.AP↗

arXiv2025

Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone $\mC$ after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling provided the cone satisfies some technique conditions. The difference from Velázquez's previous results is that we drop the symmetry condition on the cone.

math.DG↗

arXiv2024

Ancient mean curvature flows with finite total curvature

We construct an $I$-family of ancient graphical mean curvature flows over a minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the Morse index $I$ by establishing exponentially fast convergence in terms of $|x|^2-t$. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.

math.DG↗

arXiv2021

Flow by powers of the Gauss curvature in space forms

In this paper, we prove that convex hypersurfaces under the flow by powers $α>0$ of the Gauss curvature in space forms $\mathbb{N}^{n+1}(κ)$ of constant sectional curvature $κ$ $(κ=\pm 1)$ contract to a point in finite time $T^*$. Moreover, convex hypersurfaces under the flow by power $α>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in $\mathbb{N}^{n+1}(κ)$. This extends the known results in Euclidean space to space forms.

math.DG↗

arXiv2021

Ricci flow starting from an embedded closed convex surface in $\mathbb{R}^3$

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.

math.DG↗