Related papers: On the partial uniform ellipticity and complete co…
We study fully nonlinear elliptic equations on Hermitian manifolds through blow-up argument and partial uniform ellipticity. We apply our results to draw geometric conclusions on finding conformal Hermitian metrics with prescribed…
This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is…
We solve prescribed problems for modified Schouten tensors in the conformal classes of smooth complete metrics, which extends the results obtained in prequel \cite{yuan-PUE1}. The key ingredient is to confirm the uniform ellipticity of…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
In this paper, we establish a priori estimates and existence results for solutions of a general class of fully non-linear equations on noncompact K\"{a}hler and Hermitian manifolds. As geometric applications, we construct complete…
In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological…
We use certain Morse functions to construct conformal metrics with negative sectional curvature on locally conformally flat manifolds with boundary. Moreover, without conformally flatness assumption, we also construct conformal metric of…
In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator…
We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the…
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in…
This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory…
Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the…
We discuss a number of topics in the area of conformally compact Einstein metrics, mostly centered around the global existence question of finding such metrics with an arbitrarily prescribed conformal infinity. The paper is partly a survey…
This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the…
We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space…
The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy"…
We study conformally compact metrics satisfying the Lovelock equations, which generalize the Einstein equation. We show that these metrics admit polyhomogeneous expansions, thereby naturally realizing the Fefferman-Graham expansion, which…
In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature,…
We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{\"u}per, Markina and Silva Leite, which includes the well-known embedded and canonical metrics on Stiefel manifolds as special cases. The…
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.