Narrow Results

In this paper, we extend the local gradient estimates established in [8, 10] to the conformal quotient equations. An existence of solutions of the conformal quotient equations follows from the local gradient estimates and the paper [14] of Gursky and Viaclovsky.

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ($Q$-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s $Q$-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

Let $(M,𝜃)$ be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator $L_{𝜃}$. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is $n\ge 7$ and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.

In this paper, we study conformal deformations and $C$-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to $C$-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

In this note we will review recent results concerning two geometric problems associated to the scalar curvature. In the first part we will review the solution to Schoen's conjecture about the compactness of the set of solutions to the Yamabe problem. It has been discovered, in a series of three papers, that the conjecture is true if and only if the dimension is less than or equal to 24. In the second part we will discuss the connectedness of the moduli space of metrics with positive scalar curvature in dimension three. In two dimensions this was proved by Weyl in 1916. This is a geometric application of the Ricci flow with surgery and Perelman's work on Hamilton's Ricci flow.