Łojasiewicz inequalities for mean convex self-shrinkers

Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

We show that the product of two round shrinking spheres is an isolated self-shrinker in any codimension, modulo rotations. Moreover we prove explicit Łojasiewicz inequalities near such products. Łojasiewicz inequalities were previously used by Schulze to prove uniqueness of tangent mean curvature flows at compact shrinkers; our results provide an explicit rate of convergence to products of two spheres.

柱形自收缩子的Łojasiewicz不等式、唯一性与刚性

Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

该研究针对平均曲率流的一类柱形自收缩子建立Łojasiewicz不等式，证明对应奇点爆破的唯一性与这类柱体的孤立性，解答了Colding-Minicozzi的相关猜想，其直接微扰分析方法对圆柱情形也具创新性。

Estimates for eigenvalues of Lr operator on self-shrinkers

Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partialΩ$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is defined by $$\mathcal{L}_r=e^{\frac{|x|^2}{2}}{\rm div}(e^{-\frac{|x|^2}{2}}T^r\nabla\cdot)$$ with $T^r$ denoting a positive definite (0,2)-tensor field on $M$. We obtain "universal" inequalities for eigenvalues of the operator $\mathcal{L}_r$. These inequalities generalize the result of Cheng and Peng in \cite{ChengPeng2013}. Furthermore, we also consider the cas…

自收缩子的高斯加权曲率间隙

Gaussian-Weighted Curvature Gaps for Self-Shrinkers

本文利用法坐标函数和加权Poincaré不等式，证明嵌入自收缩子高斯加权L²曲率积分的下界，并结合熵间隙定理得到非球面闭自收缩子的严格曲率间隙，在二维情形推广到完整嵌入自收缩子。

$\mathbb{R}^{n+1}$ 中 $Q_k$-流的自收缩子的体积增长

Volume Growth of Self-Shrinkers of the $Q_k$-Flow in $\mathbb{R}^{n+1}$

本文建立无曲率假设的完备黎曼流形体积估计，证明欧几里得空间中一类可定向超曲面（含 $Q_k$-流自收缩子和特定极小超曲面）具有至多多项式体积增长和有限加权体积。

大规模正则性与Ricci收缩子上的Weyl定律

The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers

本文证明完备梯度Ricci收缩子具有大规模正则性，并由此推出其加权拉普拉斯算子满足经典Weyl定律。