Positive Yamabe Research Articles

Gluing theorems for constant scalar curvature manifolds

This paper concerns the positive singular Yamabe problem (SYP) in the case where the set of singularities is finite. The main result of the paper is the construction of solutions to the SYP for compact manifolds non-degenerate manifolds M that are conformally flat in a neighborhood of the singular set. We use a gluing method to construct solutions to the SYP, by attaching complete non-compact constant scalar curvature Delaunay ends to M with small balls removed.

A Kazden-Warner type condition and heat kernel estimates on noncompact manifolds

Let M be a noncompact manifold with nonnegative Ricci curvature outside a compact set and with nonnegative scalar curvature. First we generalize and extend the famous Kazdan-Warner and Bourguignon-Ezin condition, from compact manifolds to M. Second we show that the Yamabe problem of prescribing constant positive scalar curvature on those M with positive Yamabe invariant does not have a finite energy solution in most cases. We also prove qualitatively sharp two-sided large time estimates of Schrodinger heat kernels including those of the conformal Laplacian with long range scalar curvature.

Harmonic maps and the topology of conformally compact Einstein manifolds

We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

A conformally invariant classification theorem in four dimensions

In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.