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].

Hyperbolic submanifolds with finite type hyperbolic Gauss map

2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin
Keyword(s):  
Scalar Curvature ◽  
Finite Type ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Gauss Map ◽  
Hyperbolic Spaces ◽  
Nonzero Constant ◽  
Hyperbolic Gauss Map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

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.

scholarly journals Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

1972 ◽  
Vol 45 ◽  
pp. 139-165 ◽  
Author(s):  
Joseph Erbacher
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Additional Assumption ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Normal Connection ◽  
Isometric Immersions ◽  
The Mean ◽  
Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

EMBEDDED HYPERSURFACES WITH CONSTANT mTH MEAN CURVATURE IN A UNIT SPHERE

2010 ◽  
Vol 12 (06) ◽  
pp. 997-1013 ◽  
Author(s):  
GUOXIN WEI ◽  
QING-MING CHENG ◽  
HAIZHONG LI
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Scalar Curvature

In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere Sn+1(1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature Hm > 0 in Sn+1(1), for 1 ≤ m ≤ n-1. Moreover, if the 2nd mean curvature H2 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 2 and n ≥ 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H2 (i.e. constant scalar curvature) in Sn+1(1); If the 4th mean curvature H4 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 3 and n ≥ 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H4 in Sn+1(1).

Conformal Geometry Solitons

2012 ◽  
Vol 472-475 ◽  
pp. 123-126
Author(s):  
Rong Rong Cao ◽  
Xiang Gao
Keyword(s):  
Scalar Curvature ◽  
Compact Manifold ◽  
Hilbert Function ◽  
Conformal Geometry ◽  
Constant Scalar Curvature ◽  
Yamabe Flow ◽  
Noncompact Manifolds ◽  
Monotone Formula ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we deal with a generalization of the Yamabe flow named conformal geometry flow. Firstly we derive a monotone formula of the Einstein-Hilbert functional under the conformal geometry flow. Then we prove the properties that the conformal geometry solitons and conformal geometry breather both have constant scalar curvature at each time by using the modified Einstein-Hilbert function. Finally we present some properties of Yamabe solitons in compact manifold and noncompact manifolds through the equation of Yamabe soliton.

scholarly journals Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

1993 ◽  
Vol 132 ◽  
pp. 155-173
Author(s):  
Shin Kato ◽  
Shin Nayatani
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Smooth Function ◽  
Compact Manifold ◽  
Second Order ◽  
Conformal Metrics ◽  
Derivatives Of

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equationwhere ▽u denotes the Laplacian of u, defined with respect to the metric g.

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