Keaton Naff
In dimensions $n \geq 5$, we prove a canonical neighborhood theorem for the mean curvature flow of compact $n$-dimensional submanifolds in $\mathbb{R}^N$ satisfying a pinching condition $|A|^2 < c|H|^2$ for $c = \min \{ \frac{3(n+1)}{2n(n+2)},\frac{1}{n-2}\}.$
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This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The conditions that determine a Lie symmetry in such a case are completely integrated. Finally the results obtained are compared with some four-dimensional Lie groups whose Lie algebras have three-dimensional abelian ni…
In this purely expository note, we recall a few known properties of 3-dimensional singularity models. These properties are direct consequences of Perelman's canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional kappa-solutions.
A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in $S^{k+1}$. We develop a new proof strategy which can also obtain an analogous result for codimension 2 compact minimal submanifolds of $S^{k+1}$.
Curvature property on Hermitian Lie algebras with abelian ideals of codimension two
本文研究含余维二交换理想的单模Hermitian李代数，证明常Chern全纯截面曲率蕴含Chern平坦，且其他典范联络的常全纯截面曲率导致Kähler平坦及商代数的交换性。
On $κ$-solutions and canonical neighborhoods in 4d Ricci flow
本研究提出四维Ricci流中$\kappa$-解的分类猜想，构造了新的对称单参数族解，建立了穿过柱形奇点的典范邻域定理，并指出商颈带来的新现象与非唯一性示例。
In this paper, we prove several formulas related to Hodge theory, and using them to prove the deformations of a compact $H$-twisted generalized Calabi-Yau manifold are unobstructed and $L^2$ convergence in a neighborhood in another power series . And if we assume that the deformation is smooth in a fixed neighborhood, and assume the existence of a global canonical family of deformation, we also construct the global canonical family of the deformations of generalized Kähler manifolds.
The canonical submersion of $\mathcal{S}$-manifolds and transverse Kähler-Einstein structures
本文研究带特征联络的$\u0002mathcal{S}$-流形的和乐性质，证明其和乐的特殊性质，建立局部黎曼淹没下的曲率与Einstein条件双射，推广Sasakian对应至任意CR余维，并给出Ricci平坦的等价刻画。
We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^2$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic c…
On the existence of canonical multi-phase free boundary Brakke flows: a case study
研究上半空间中从可数n-可整流集出发的余维1典型多相自由边界Brakke流的全局存在性，在初始数据满足均匀密度比假设下，证明自由边界在短时间内无正质量。
We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain symmetry class the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode cons…
We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension $1$ symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finite-dimensional family of solutions. The results are new for Ricci flow.
Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
本文研究均凸邻域猜想，证明在所有维度和圆柱形奇点类型下，均值曲率流在奇点附近均凸，并建立局部模型定理。同时，通过分类古老渐近圆柱形流，提出更统一的猜想版本。
Canonical foliation of bubblesheets
本文提出准平行平均曲率（QPMC）这一新曲率条件，并证明在接近标准度量的乘积流形上存在由QPMC球面构成的典范叶状结构，从而为几何流中的气泡片建立标准形式。
A symplectic slice theorem
我们提供了一个模型，在辛流形中的任何轨道的一个开放的不变的邻域被赋予一个典型的正常对称。我们的结果推广了马尔和Guillemin和Sternberg的典型对称性，有一个相关的动量映射的建设。在这些论文中，动量图在管状模型的构建中发挥了至关重要的作用。目前的工作表明，在管状模型的建设，它可以使用所谓的朱映射代替，它存在于任何典型的行动，不像动量映射。任何不变哈密顿函数的哈密顿方程在这些管状变量中呈现出特别简单的形式。作为一个应用，我们将发现的情况下，我们将呼吁tubewise哈密顿，其中存在的一个标准的动量映射在不变的邻域是保证。
On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially explicit e-structure of the concerned local biholomorphic equivalence problem. More advanced explorations of the incoming differential invariants due to the first and to the third authors already appeared in January 2014, hence the purpose is to show, while studying this specific Cla…
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their u…
This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise…
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