Geometric realizations of curvature models by manifolds with constant scalar curvature

We show any pseudo-Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and -scalar curvature.     

K-stability of constant scalar curvature Kähler manifolds

We show that a polarised manifold with a constant scalar curvature Kähler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S.K. Donaldson.     

Complete and non-compact conformally flat manifolds with constant scalar curvature

Non-compact conformally flat manifolds with constant scalar curvature and non-compact Kaehler manifolds with vanishing Bochner curvature are studied and classified.Partially supported by TGRC-KOSEF, 1990.     

Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big (M^n,\,g,\,X,\,\lambda \big )$ with constant scalar curvature is isometric to a Euclidean sphere $\mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.     

Blowing up and desingularizing constant scalar curvature Kähler manifolds

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

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

Classification of pinched positive scalar curvature manifolds

Let (Mn,g), n?3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that Rg?n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition

     

Prescribed scalar curvature problem on complete manifolds

Conditions on the geometric structure of a complete Riemannian manifold are given to solve the prescribed scalar curvature problem. In some cases, the conformal metric obtained is complete. A minimizing sequence is constructed which converges strongly to a solution.     

Conformal scalar curvature equations on complete manifolds

This note contains considerations on the existence and non-existence problem of conformal scalar curvature equations on some complete manifolds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radius. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conformally compact manifolds. We obtain non-existence results on both types of conditions. Then we study in more details the existence problem on manifolds with cones, manifolds with cusps and conformally flat manifolds of bounded positive scalar curvature.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997