Jeffrey Galkowski, Steve Zelditch
We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.
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In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
On the existence and stability of blowup for wave maps into a negatively curved target
考虑$(1+d)$维Minkowski空间上的波映射.对于每一个维度$d\geq 8$，我们构建一个负弯曲，$d$维目标流形，允许存在一个自相似波映射，提供了一个稳定的爆破机制，相应的柯西问题。
Nonuniqueness of solutions of the Navier-Stokes equations on negatively curved Riemannian manifolds
在一个著名的工作，M。Anderson构造了一个Hadamard流形$(M^n, g)$，当$p \neq n/2$时，它带有非零的$L^2$调和$p$ -形式，从而反驳了Dodziuk-Singer猜想.本文利用流形$(M^3, g)$解决了几何分析中的另一个问题，即Navier-斯托克斯方程的Leray-Hopf型解的非唯一性.
Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces
研究了紧致负曲面上的半经典薛定谔方程。对于任何序列的初始数据microlocalized上的单位余切丛，我们看看量子演化（以下的Eschlafest时间）小扰动下的薛定谔方程，我们证明，在半经典极限和典型的扰动，解决方案成为均匀分布的单位余切丛。
Integral Formula for the Characteristic Cauchy Problem on a curved Background
本文利用Friedlander在他的著作《弯曲时空上的波动方程》（1975）中提出的方法，给出了具有任意自旋的Dirac方程的特征Cauchy问题的一个局部积分公式，该公式适用于一般的弯曲时空。Penrose在“广义相对论中任意自旋经典场的超曲面初始数据”（Gen.Rel. 1980年（昭和10年），被撤销。我们期望这种方法可以用来得到Dirac方程特征Cauchy问题的精确估计。
Long-time dynamics of the perturbed Schrödinger equation on negatively curved surfaces
研究了具有常负曲率的无边界紧致黎曼曲面上半经典Schr{<$}dinger方程的扰动.我们表明，对于尺度的时间是对数的扰动的大小，在一个小的光谱窗口中的初始数据的解决方案成为equidistributed在半经典的限制。作为该方法的一个应用，我们还导出了小光谱窗口中初始数据在Escherfest时间以下和之后的量子Loscherfest回波的一些性质。
Small scale quantum ergodicity in negatively curved manifolds
主要定理已被稍微推广，以包括更大类的符号。
Global Solutions of Semilinear Parabolic Equations on Negatively Curved Riemannian Manifolds
研究了具有负截面曲率的黎曼流形上半线性抛物型方程解的整体存在性。一个特别注意的是一类初始条件，确保整体解的存在性。事实上，我们表明，这样一个类是至关重要的曲率界。
Dispersion and controllability for the Schrödinger equation on negatively curved manifolds
研究了紧致黎曼流形上的含时薛定谔方程$ \imath\frac{\partial u}{\partial t}=-1/2Δu,$，在该流形上的测地流具有Anosov性质.利用半经典测度的概念，我们证明了与薛定谔传播子的色散性质和可控性有关的各种结果。
Constant potentials do not minimize the fundamental gap on convex domains in negatively curved Hadamard manifolds
该研究证明在负曲率流形中，常数势不最小化基本间隙，揭示了基本间隙猜想第二部分的错误。
The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model $n$-dimensional hyperbolic space, complementing the results of Borisov and Freitas (\textit{Comm. Anal. Geom.} 25: 507--544, 2017). We then give a synthetic proof of Cheng's sharp eigenvalue comparison theorem in metric measure spaces satisfying a 'negatively curved' Bishop-Gromov-type volume monotonicity hypothesis. A…
Ricci flow of negatively curved incomplete surfaces
我们证明了从一致负曲率曲面出发的里奇流的唯一性，并假设流动是瞬时完全的。与[10]中证明的更一般的存在性结果一起，这解决了该类中的适定性问题。
This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a que…
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into $\mathbb{R}^3$. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a $C^{1,1}$ isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in $L^2$.
Isometric embedding of negatively curved complete surfaces in Lorentz-Minkowski space
Hilbert-Efimov定理指出，任何曲率上界为负常数的完备曲面不能等距嵌入到$\mathbb{R}^3.$中。本文证明了任何曲率上界为负常数的单连通光滑完备曲面都允许等距光滑嵌入到Lorentz-Minkowski空间$\mathbb{R}^{2,1}$中。
We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These include the standard hyperbolic plane, generalised helicoid-type metrics and generalised Enneper metrics. Our proof is based on the method of compensated compactness and invariant regions in hyperbolic conservation laws, together with several observations on the geometric quantities (Gauss curvature, metric components etc.) of negatively curved surfaces.
Non-negatively curved Kähler manifolds with average quadratic curvature decay
设$(M, g)$是一个完备的非紧Kähler流形，它具有非负的有界全纯双截曲率.推广我们在\cite{CT3}中发展的技巧，我们证明了$M$的泛覆盖$\wt M$是$\ce^n$的双全纯，只要$(M, g)$具有平均二次曲率衰减，或者$M$支持具有非负和一致有界全纯双截曲率的Kähler-里奇流的永恒解.我们还分类某些局部限制所产生的Kähler-里奇流的注入半径的统一估计的情况下。
Negatively Curved Minimal Surfaces in the Round Four-Sphere
该研究在四维单位球面中构造了处处负高斯曲率的光滑闭嵌入极小曲面，其亏格随整数m增大而增大，面积趋近于4π²，解决了丘成桐1982年公开问题列表中的第101个问题。
Nonexistence of invariant nodal line and improved $L^2$ restriction bounds for Neumann data on negatively curved surface
本文针对负曲率曲面上的拉普拉斯特征函数限制问题，证明了不变节线不存在，并获得了 Neumann 数据 L^2 限制下界的定量改进。
The porous medium equation with measure data on negatively curved Riemannian manifolds
研究负弯曲黎曼流形上多孔介质方程柯西问题弱解的存在性和唯一性。我们证明了解的存在性，作为初始条件的有限氡措施，不一定是积极的。然后，我们建立在类的非负解的唯一性。另一方面，我们证明了多孔介质方程的任何弱解都必须以有限Radon测度为初始基准。此外，我们还得到了流形上势分析中的一些结果，涉及到上调和函数的中值不等式的修正形式的有效性，以及正Radon测度的势的性质.这样的结果是新的和独立的利益，是至关重要的，我们的方法。
Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds
本文在负曲率紧流形上改进薛定谔方程的Strichartz估计，在双曲曲面上得到与Ehrenfest时间尺度对应的无损失估计，并给出非正曲率流形上球面无法成立的改进端点估计。
This paper is devoted to several small data existence results for semi-linear wave equations on negatively curved Riemannian manifolds. We provide a simple and geometric proof of small data global existence for any power $p\in (1, 1+\frac{4}{n-1}]$ for the shifted wave equation on hyperbolic space ${\mathbb H}^n$ involving nonlinearities of the form $\pm |u|^p$ or $\pm|u|^{p-1}u$. It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru) on Euclidean space. We also prove a small data existence theorem for variably curved backgrounds which extends earlier ones for …
Global minimizers for fast diffusion versus nonlocal interactions on negatively curved manifolds
本文在Cartan-Hadamard流形上研究含快速扩散与非局部相互作用的自由能泛函，给出全局极小化子存在的充要条件，并推导新的Carlson-Levin型不等式。
Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets $D$ in $M$ for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$, and the supremum of the torsion function for $D$ are comparable with the square of the capacitary width for $D$ if the latter is suffic…
The Cauchy Problem for Wave Maps on a Curved Background
本文考虑波映射u：\R乘M \to N的Cauchy问题，其中u是黎曼流形的波映射，（M，g）和（N，h）.我们证明了全局存在性和唯一性的初始数据是小的临界Sobolev范数的情况下（M，g）=（\R^4，g），其中g是一个小扰动的欧几里德度量。证明遵循Statah和Struwe介绍的方法，用于证明小数据波映射u的全局存在性和唯一性：\R \times \R^d \to N在临界范数下，d至少为4。在我们的论证中，我们采用了Metcalfe和Tataru所建立的变系数波动方程的Eschenhartz估计。
Topology of closed asymptotic curves on negatively curved surfaces
受Nirenberg问题启发，研究负曲率曲面上闭渐近曲线的拓扑，利用Călugăreanu定理给出环绕数公式，并推广了相关观察，构造了法向单射但环绕数非零的例子。
Tensor tomography on negatively curved manifolds of low regularity
本文研究了低正则性负曲率流形上的张量层析成像问题，证明了 $C^{1,1}$ 度量简单 Riemannian 流形上测地 X 射线变换的螺线管式内射性，核心方法是利用非光滑单位球丛上输运方程的 Pestov 能量估计并对光滑证明方法进行实质性修改。
We study the Cauchy problem for Schrödinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.
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