arXiv subjects
Publications and source records attributed to Yongheng Han.
The Calderón-Zygmund inequality is a cornerstone of harmonic analysis and partial differential equations. In this article, we establish various Calderón-Zygmund inequalities on evolving Riemannian manifolds with bounded curvature. We also provide concrete applications of such inequalities.
In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the continuous dependence of mean curvature flow on initial data.
We establish the $L^p$-boundedness of the local covariant Riesz transform for differential forms on manifold $M$ with bounded $\|Rm\|$. Let $Δ_j$ be the Hodge Laplace operator on $j$-forms. For any $p \in (1, \infty)$ and $κ>κ_0$, we show that the operator $\nabla (Δ_j + κ)^{-1/2}$ is bounded on $L^p(M)$. Consequently, we obtain Calderón-Zygmund estimates for manifolds with bounded Riemannian curvature.
For $n\geq 2$, we construct $I$-dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface $Σ$ with finite total curvature in $\mathbb{R}^{n+1}$, where $I$ is the Morse index of the Jacobi operator on $Σ$.
We study the multiplicity of the singularities of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.