Halfspace type theorems for self-shrinkers in arbitrary codimension

任意余维数中 Ricci 受控的完备自收缩子的刚性

Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension

本文证明任意余维数中满足 Ricci 下界条件的完备自收缩子必为线性子空间或圆球，并给出余维数二时的尖锐刚性定理。

渐近锥形高余维自收缩子与自扩张子的唯一性

Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

本文证明了渐近于同一锥的高余维自收缩子唯一且以单位重数收敛，并给出自扩张子在此条件下像重合的唯一性结果。

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.

$\mathbb R^4$ 中一个稳定的自收缩 Möbius 丛

A stable self-shrinking Möbius bundle in $\mathbb R^4$

本文证明了 $\mathbb R^4$ 中具有 Möbius 丛拓扑的嵌入平均曲率流收缩子是线性稳定的，提供了高余维数中非平坦、非球形的稳定收缩子，并揭示了稳定奇点可能编码非平凡拓扑操作。

The Rigidity Theorems of Self Shrinkers via Gauss maps

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1 and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmanian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.

A Bernstein type result for graphical self-shrinkers in $\mathbb{R}^4$

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural conditions on the Jacobian of any smooth map from $\mathbb{R}^2$ to $\mathbb{R}^2$, we show that the self-shrinker which is the graph of this map must be affine linear. The proof relies on the derivation of structure equations of graphical self-shrinkers in terms of the parallel form…

Gaussian Harmonic Forms and Two-Dimensional Self-Shrinkers

We consider 2-dimensional orientable self-shrinkers $Σ$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms minimizing the norm $\int_Σ\eterm |ω|^2$ in their cohomology class. Any closed form satisfying the Euler-Lagrange equation for this minimization will be called a Gaussian Harmonic one Form (GHF). We then use these forms to show that if such a $Σ$ has genus $\geq 1,$ then we have a lower bound on the supremum norm of $A^2$. GHF's may also be applied to create an upperbound for the lowest eigenvalue of the operator $L$. In the codi…

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。

平均曲率流闭自收缩子的一个间隙定理

A gap Theorem on closed self-shrinkers of mean curvature flow

本文针对平均曲率流的任意余维数 n 维闭自收缩子建立第二基本形式拼挤条件，证明满足该界时必为标准球面，为相关开放问题提供证据。

自收缩子的 Li-Yau 共形体积与 Colding-Minicozzi 熵

Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

本文利用自收缩子的归一化 Li-Yau 共形体积给出不依赖余维且对平面尖锐的 Colding-Minicozzi 熵下界，并借此验证实射影平面情形的相关猜想。

高斯空间中的完备 λ-子流形

Complete $λ$-submanifolds in Gauss spaces

本文研究高斯空间中任意余维数的 λ-子流形，利用散度型定理和 Simons 型恒等式证明半空间型定理与间隙定理，将自收缩子和 λ-超曲面的相关结果推广到 λ-子流形。

A Bernstein type theorem for graphic self-shrinkers with flat normal bundle

In this note we will prove that an $n$ dimensional graphic self-shrinker in $R^{n+m}$ with flat normal bundle is a linear subspace. This result is a generalization of the corresponding result of Lu Wang in codimension one case.

任意余维数中具有正常数量曲率和\\(S\le 1\\)的完备自收缩子

Complete Self-Shrinkers in Arbitrary Codimension with Positive Constant Scalar Curvature and $S\le 1$

本文证明任意余维数中具有正常数量曲率且第二基本形式平方范数不超过1的完备自收缩子必为球面或广义柱面，排除了新的高余维数例子。

Complexity of parabolic systems

We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. Combined with \cite{CM12}, this gives the first general bounds on generic singularities of surfaces in arbitrary codimension. We also show sharp bounds for codimension in arguably some of the most important situations of ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cyli…

平均曲率流中自收缩子的尖锐熵下界

Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

本文研究了平均曲率流中具有非平凡同调的自收缩子的几何不等式问题，证明了此类自收缩子的熵具有由圆球面熵给出的尖锐下界，并刻画了等号成立的条件。

$\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow

In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's results.

A rigidity theorem of $ξ$-submanifolds in $\mathbb{C}^{2}$

In this paper, we first introduce the concept of $ξ$-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $λ$-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian $ξ$-submanifold in the complex $2$-plane $\bbc^2$.

受迫平均曲率流爆破的唯一性

Uniqueness of blowups for forced mean curvature flow

本文证明了受迫平均曲率流在闭自收缩子和圆圆柱自收缩子处切锥的唯一性，推广了Schulze和Colding-Minicozzi的结果，并涵盖黎曼流形情形。