Some Uniforme Estimates for Scalar Curvature Type Equations

Critical metrics of the $L^2$-norm of the scalar curvature

In this paper we investigate complete critical metrics of the $L^{2}$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.

Remarks on scalar curvature of Yamabe solitons

In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of a complete non-compact Yamabe soliton has non-negative scalar curvature. A new proof of Kazdan-Warner condition is also presented.

The existence results for solutions of indefinite scalar curvature problem

In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.

Blowing up and desingularizing constant scalar curvature Kähler manifolds

This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kaehler metrics.

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.

Prescribing the scalar curvature in the null case

We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.

Scalar curvature flow on S^n to a prescribed sign-changing function

In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function $f$ as its scalar curvature.

三维空间中预定数量曲率超曲面的内部曲率估计

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

本文通过积分方法，证明了三维空间中预定数量曲率方程超曲面的内部曲率估计，关键发现是满足该方程的图函数对应的拉格朗日子流形具有有界平均曲率。

任意维度下图形标量曲率方程的内部曲率估计

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。

The Scalar Curvature Equation on $S^3$

We give existence results for solutions of the prescribed scalar curvature equation on $S^3$, when the curvature function is a positive Morse function and satisfies an index-count condition.

Singular scalar curvature equations

We develop estimates for the equation of scalar curvature of singular metrics with cone angle $β>1$, in a big and semi-positive cohomology class on a Kähler manifold. We further derive the Laplacian estimate for the scalar curvature equation of degenerate Kähler metrics. We then have several applications of these estimates on the singular constant scalar curvature Kähler metrics, which also include the singular Kähler-Einstein metrics.

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of "lower energy" solutions to the above equation fails when the dimension of the manifold is not less than $62$.

A Uniform Estimate for Scalar Curvature Equation on Manifolds of dimension 4

On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.

Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature --- Part II

In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $\frac12$-Hölder continuous in a uniform sense. This implies that the distance function can be extended continuously up to the singular time.

The Kahler Metrics of constant scalar curvature on the Complex Torus

We study the Dirichlet problem of the Abreu equation. The solutions provide the Kahler metrics of constant scalar curvature on the complex torus.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+\infty$.

Conformal Scalar Curvature Equations in Open Spaces

The article contains a brief description on the study of conformal scalar curvature equations, and discusses selected topics and questions concerning the equations in open spaces.

A priori estimates for the scalar curvature equation on $S^3$

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on $S^3$. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse function.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $σ_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition.

Prescribing scalar and boundary mean curvature on the three dimensional sphere

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

A prescribed scalar and boundary mean curvature problem on compact manifolds with boundary

We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.

Some existence results for the Webster scalar curvature problem in presence of symmetry

We prove some existence results for the Webster scalar curvature problem on the Heisenberg group and on the unit sphere of ${\mathbb C}^{n+1}$, under the assumption of some natural symmetries of the prescribed curvatures. We use variational and perturbation techniques.

On the Webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

Hypersurfaces with nonnegative scalar curvature

We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relies on a new geometric inequality which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete non-compact hypersurfaces of finitely many regular ends with nonn…

标量曲率的通用波纳切公式

A universal Bochner formula for scalar curvature

本文提出了一种适用于标量曲率的通用波纳切公式，涵盖了最小切片稳定性不等式、施罗德inger-利钦纳沃伊兹型公式及高维斯特恩水平集恒等式的特殊情况。

An End to End Gluing Construction for Metrics of Constant Positive Scalar Curvature

Complete constant positive scalar curvature metrics on S^n - {p_1, ..., p_k} admit a definite asymptotic structure; i.e. the metric is asymptotic to a specific S^{n-1}-invariant metric near the puncture points. This allows one to glue together two such metrics near their puncture points, provided the asymptotic structures match, and create a metric whose scalar curvature is very close to constant. In this paper we perturb this new metric to obtain a metric of constant scalar curvature.

The Scalar Curvature Problem on the Four Dimensional Half Sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.

Prescribed non positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation

We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.

深度1轨形奇点解消上的常标量曲率Kähler度量

Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1

本文针对沿高复余维子轨形具有I型奇点的常标量曲率Kähler轨形，构造其适当解消上的常标量曲率Kähler度量新例。

The Webster scalar curvature flow on CR sphere. Part II

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.

无穷远处的链接与非紧致流形上的标量曲率衰减

Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds

研究非紧致正标量曲率流形上曲率衰减与无穷远处拓扑结构的关系，证明拓扑链接迫使弱有界几何流形上标量曲率多项式衰减，并发展局部于非紧致流形端点的障碍理论。

Estimates for metrics of constant Chern scalar curvature

We prove a priori estimates for constant Chern scalar curvature metrics on a compact complex manifold conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the Kähler setting.

Scalar curvature rigidity of geodesic balls in S^n

In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).

Scalar curvature, mean curvature and harmonic maps to the circle

We study harmonic maps from a 3-manifold with boundary to $\mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $π/ 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.