Matthias Schneider
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.
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A priori interior estimates for special Lagrangian curvature equations
本文针对特殊拉格朗日曲率方程，在临界相和凸情形下建立了先验内部曲率估计，并证明了任意常数相的先验内部梯度估计。
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。
A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase
本文针对超临界相拉格朗日平均曲率流，证明奇点的内部先验估计，建立临界及超临界下成立的雅可比不等式，并推广至更广的拉格朗日平均曲率型方程。
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.
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.
We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case $n=2$ and positive curvature, and then by P. Guan and the first author for non-negative curvature and $n=2$. Using $C^{2,α}$ interior estimates of Evans and Krylov for concave fully nonlinear elliptic partia…
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.
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.
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.
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.
We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.
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.
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。
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。
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.
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.
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 $62$.
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.
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.
We study the Dirichlet problem of the Abreu equation. The solutions provide the Kahler metrics of constant scalar curvature on the complex torus.
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 $+\infty$.
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.
Interior estimates and regularity for the scalar curvature equation in dimension 4
该研究针对四维标量曲率方程，采用Shankar和Yuan[SY25]的加倍法，结合小梯度假设的先验加倍不等式与新迭代旋转论证，建立了内部曲率估计及C¹粘性解的内部正则性。
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.
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.
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.
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.
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.
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…
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。
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an in…
The regularity theory of the degenerate complex Monge-Ampère equation is studied. The equation is considered on a closed compact Kähler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a solution $ϕ$ of the degenerate complex Monge-Ampère equation $\det(g_{i \bar{j}} + ϕ_{i \bar{j}}) = f \det(g_{i \bar{j}})$, it is shown that the Laplacian of $ϕ$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and $\inf_M Δf^{1/(m-1)}$.
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.
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。
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.
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。
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.
We give some estimate of type sup*inf for scalar curvature type equations.
This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results. These methods were first introduced by A. Bahri.
We give some estimates of type sup*inf for the prescribed scalar curvature equation in dimension 4 and 5, under some condtion on the prescribed curvature.
Harnack inequalities for equations of type prescribed scalar curvature
本文研究标量曲率类型方程在n≥4维空间中解的Harnack不等式，提出新的不等式形式并分析其应用条件。
Using the Lyapunov-Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S^n, (n > 5), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C^{n - 1, β} smoothness.
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]).
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.
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