Aobing Li, Huan Zhu
In this paper, we consider a fully nonlinear problem on manifolds with boundaries of negative admissible curvatures. As a consequence, we conclude the existence of certain types of metrics on the general differential manifolds with boundaries.
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We give sufficient conditions on a function invariant under the action of an isometry group to be Branson's Q-curvature of a metric in a given conformal class, using the conformal GJMS operators.
Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
该研究解决了长期问题，证明了所有维度n≥3的常数图形标量曲率方程容许解的内部曲率估计，其证明结合雅可比不等式、双曲面最大值原理与双曲面波戈列洛夫估计。
In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $σ_2(κ(X))=f(X, ν(X)).$ In this note, we give a simpler proof of this result, and extend it to space forms.
Prescribed $L_p$ quotient curvature problem and related eigenvalue problem
本文研究预定 $L_p$ 商型曲率问题的可容许光滑解存在性，针对 $p$ 与 $k-l+1$ 的不同关系给出解的存在性结果，并在 $p<k-l+1$ 情形下证明偶函数条件是必要的。
Interior gradient estimates for prescribed curvature equations in hyperbolic space
该论文聚焦数学分析领域，针对双曲空间内的预定曲率方程，开展其容许解的内部梯度估计相关研究。
Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature
本文研究了闭合负曲率流形上基于修改Schouten张量的非线性给定曲率问题，证明在满足特定结构条件时，此类问题总能被解决。
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric
We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.
We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.
Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.
Prescribing negative curvature with cusps and conical singularities on compact surface
本研究针对紧曲面，证明了一类仅在有限个带指定角度的尖点或锥形奇点处例外、曲率由负函数指定的共形度量的存在性与唯一性。
A priori estimates for negative constant scalar curvature conformal metrics with positive constant boundary mean curvature
针对带边界紧致黎曼流形上内部负常数数量曲率、边界正常数平均曲率的共形度量集合，在Yamabe共形不变量为正的条件下，证明其在三维及任意≥3维局部共形平坦且边界脐点的情形下先验有界。
Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $M$ supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kähler-Ricci flow in the absence of uniform estimates on the injectivity radius.
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M$ contains no smooth, complete immersed surface with curvature below $K_1$. Optimal values of $K_1$ are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Amp{è}re equations.
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other in…
We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold $(M^n,g)$, and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever $M^n$ has even a single tangent plane of negative sectional curvature. In particular, the fundamental gap conjecture strongly fails for small deformations of Euclidean space which introduce any negative curvature. We also show that when the curvature is negatively pinched, it is possible to construct such domains of any diameter up to the diameter of …
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}.
Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature
本文通过共形变形在对称圆盘上规定负高斯曲率及边界测地曲率，采用变分方法结合单调性技巧、爆破分析和Morse指标证明存在性，并给出不存在性结果。
Conformal Metrics on the Disk with Prescribed Negative Gaussian Curvature and Boundary Geodesic Curvature
本文研究了通过共形变换给定圆盘负高斯曲率和边界测地曲率的问题，通过变分方法和发散分析得出存在性结果。
The Dirichlet problem for the prescribed curvature equations in Minkowski space
针对Minkowski空间中预定曲率类空超曲面，本文研究其Dirichlet问题，并为对应完全非线性椭圆PDE的容许解建立新的内部二阶估计。
Interior curvature estimates for graphical curvature quotient equations
本文在完整Gårding锥上证明了曲率商方程可容许图形解的内曲率估计，无需凸性假设，通过压缩不等式、双曲面比较和Pogorelov估计实现。
A Comparison Pogorelov Estimate for Graphical $σ_k$-Curvature Equations
为图形σ_k-曲率方程的可容许解建立比较Pogorelov估计，推广Qiu-Yan两曲面估计，覆盖k≥2及n/2≤k<n情形，证明结合两曲面定位、混合Gårding比较与谱凹性不等式。
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
We study the Gauss curvature equation with negative singularities. For a local mean field type equation with only one negative index we prove a uniqueness property. For a global equation with one or two negative indexes we prove the non-degeneracy of the linearized equations.
In this paper, we consider fully nonlinear equations of Krylov type on Riemannian manifolds with negative curvature which naturally arise in conformal geometry. Moreover, we prove the a priori estimates for solutions to these equations and establish the existence results. Our results can be viewed as an extension of previous results given by Gursky-Viaclovsky and Li-Sheng.
In this paper, we study the Ma-Trudinger-Wang (MTW) conditions for cost functions $c$ which are of the form $c=l\circ d$, where $d$ is a Riemannian distance function with constant sectional curvature. In this case, the MTW conditions are equivalent to some computable conditions on the function $l$. As a corollary, we give some new costs on Riemannian manifolds of constant negative curvature for which the MTW conditions are satisfied.
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