Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

双曲混合流形中极大测地子流形的有限性

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

该研究证明大类非算术双曲n-流形（含Gromov等的混合流形）的有限体积浸入全测地超曲面有限，且高余维下维数≥2的极大测地子流形有限，证明结合算术群结构、动力系统与负曲率几何。

凸核中全测地子流形的正常性与有限性

Properness and finiteness of totally geodesic submanifolds in the convex core

本文研究几何有限秩一局部对称流形凸核内的全测地子流形，证明其凸核内维数≥2的极大全测地子流形正常浸入、体积有限且数量有限，结合已有工作推导得出含无穷多该类子流形的对应流形必为算术流形。

趋于哈密顿稳态子流形的几何流

A Geometric flow towards hamiltonian stationary submanifolds

本文提出一种Kähler流形中Lagrangian子流形的几何流，保持初始Hamiltonian同痕类且为体积梯度流，其稳态解为Hamiltonian稳态子流形。通过Weinstein邻域定理转化为四阶严格抛物型方程，并证明了短时存在性、唯一性及高阶估计。

C^{1，1}黎曼子流形的厚度公式和C^1- Compactness

Thickness Formula and C^1-Compactness for C^{1,1} Riemannian Submanifolds

利用几何焦距和双临界点证明了完备黎曼流形M^n的C^{1，1}紧子流形K^k的法内射性半径（厚度）i（K，M）的一个公式.我们还证明了包含在固定流形M的紧致子集D中的所有紧致子流形K的集合的C^1紧性，其中i（K，M）有界远离0。这是一个与Gromov紧性定理有关的外生等距嵌入型定理。这就产生了在一个合痕类中存在最厚的子流形，其中K不一定是连通的。

Topological and physical knot theory are distinct

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.

Curvature of the space of positive Lagrangians

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an…

拉格朗日双球面可以辛纽结

Lagrangian two-spheres can be symplectically knotted

本文证明了辛四流形M具有以下性质：M中的光滑嵌入双球面的一个合痕类包含无穷多个拉格朗日子流形，其中没有两个子流形是同素的拉格朗日子流形.这些例子是使用一类特殊的辛自同构（“广义德恩扭曲”）构造的。证明使用了弗洛尔同调。修订版本：删除一个脚注，增加一个参考资料

A Reifenberg type characterization for $m$-dimensional $C^1$-submanifolds of $\mathbb R^n$

We provide a Reifenberg type characterization for $m$-dimensional $C^1$-submanifolds of $\mathbb R^n$. This characterization is also equivalent to Reifenberg-flatness with vanishing constant combined with suitably converging approximating $m$-planes. Moreover, a sufficient condition can be given by the finiteness of the integral of the quotient of $θ(r)$-numbers and the scale $r$, and examples are presented to show that this last condition is not necessary.

$L^2$-harmonic $p$-forms on submanifolds with finite total curvature

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is less than a positive constant depending only on $n$. Second, we show that the dimension of $H^p(L^2(M))$ is finite if the total curvature of $M$ is finite. The vanishing theorem is a generalized version of Gan-Zhu-Fang theorem and the finiteness theorem is an extension of Zhu-Fang theorem.

同调 Lagrangian 单值性与 Wang 正合序列

Homological Lagrangian monodromy and Wang exact sequence

该研究利用能量阈值条件证明 Lagrangian 子流形的同调单值性在无低能 J-全纯球面或圆盘时为平凡。

Positive solutions to Schrödinger equations and geometric applications

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds …

几何结理论中Palais-Smale条件的探讨

On the Palais-Smale condition in geometric knot theory

本文研究了几何结理论中多种能量函数在弧长参数化结子流形上满足Palais-Smale条件，并证明了这些能量存在最小化结及临界结的光滑性。