相关论文: Thinness for Scalar-Negative Singular Yamabe Metri…

In this paper, we rigorously analyze the scalar curvature of complete expanding gradient Yamabe solitons. We completely classify nontrivial complete expanding gradient Yamabe solitons in both cases: when the scalar curvature is greater than…

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also…

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions…

We develop estimates for the equation of scalar curvature of singular metrics with cone angle $\beta>1$, in a big and semi-positive cohomology class on a K\"ahler manifold. We further derive the Laplacian estimate for the scalar curvature…

A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded…

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. Moreover, this result may be localized to compact subdomains in…

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal…

We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced…

We study new classes of metric transformations in the context of scalar-tensor theories, which involve both higher derivatives of the scalar field and derivatives of the metric itself. In general, such transformations are not invertible as…

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the…

We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature…

We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides time-symmetric initial data sets for the vacuum Einstein equations…

In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended…

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an…

We analyze the problem of the existing ambiguities in the conformal anomaly in theories with external scalar field in curved backgrounds. In particular, we consider the anomaly of self-interacting massive scalar field theory and of Yukawa…

We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for…

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…

This paper addresses the quantitative stability for a Yamabe-type functional on compact manifolds with boundary introduced by Escobar. Minimizers of the functional correspond to scalar-flat metrics with constant mean curvature on the…

The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal K\"ahler. The critical points of the total holomorphic curvature and total Ricci…

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a…