In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M, g) is compact. Established in the locally conformally flat case by Schoen [43, 44] and for n ≤ 24 by Khuri– Marques–Schoen [26], it has revealed to be generally false for n ≥ 25 as shown by Brendle [8] and Brendle–Marques [9]. A stronger version of it...

Let R be a constant. Let Mγ be the space of smooth metrics g on a given compact manifold Ω (n ≥ 3) with smooth boundary Σ such that g has constant scalar curvature R and g|Σ is a fixed metric γ on Σ. Let V (g) be the volume of g ∈ Mγ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V (·) in Mγ .

In this paper, we obtain the isoperimetric inequality on a conformally flat manifold with finite total Q-curvature. This is a higher dimensional analog of Li and Tam’s result [“Complete surfaces with finite total curvature.” Journal of Differential Geometry 33 (1991): 139–68] on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconf...

According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold M which is compatible with the canonical flat conformal structure. He showed that this metric gN has a large symmetry if gN is a complete metric. Under certain assumptions including the completeness of gN , the isomet...

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an Abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound conformally flat structures in a conformally invariant way. The eta invariant ...

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive. Dedicated to the memory of Profess...

Usual inflation is realized with a slow rolling scalar field minimally coupled to gravity. In contrast, we consider dynamics of a scalar with a flat effective potential, conformally coupled to gravity. Surprisingly, it contains an attractor inflationary solution with the rapidly rolling inflaton field. We discuss models with the conformal inflaton with a flat potential (including hybrid inflati...

This work delves into the space-time theory of 4-dimensional Kaehler manifold. Since isotropic pressure, energy density, and momentum tensor all vanish in a perfect fluid manifold, we have established that this manifold is an Einstein studied equation with cosmological constant it. Finally, demonstrated that, on conformally flat, perfectly velocity vector field infinitesimally spatially isotrop...

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...