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

2020 ◽  
Vol 269 (11) ◽  
pp. 9432-9470
Author(s):  
Xuezhang Chen ◽  
Nan Wu
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Blow Up ◽  
Constant Scalar Curvature ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Boundary Mean Curvature

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

2019 ◽  
Vol 21 (03) ◽  
pp. 1850021 ◽  
Author(s):  
Xuezhang Chen ◽  
Liming Sun
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Constant Scalar Curvature ◽  
Riemannian Metric ◽  
Conformal Metrics ◽  
Nonzero Constant ◽  
Dimension Formula ◽  
Yamabe Constant ◽  
Boundary Mean Curvature

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 [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] 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 [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Generalized -Einstein Real Hypersurfaces in and

2020 ◽  
Vol 63 (4) ◽  
pp. 909-920
Author(s):  
Yaning Wang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Coefficient ◽  
Real Hypersurface ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Hypersurfaces

AbstractIn this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

Gradient estimates of mean curvature equation with Neumann boundary condition in domains of Riemannian manifold

2017 ◽  
Vol 28 (08) ◽  
pp. 1750065
Author(s):  
Jinju Xu ◽  
Dekai Zhang
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Existence Result ◽  
Neumann Boundary ◽  
Gradient Estimates ◽  
The Maximum Principle ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation

We study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.

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