Related papers: The mixed Yamabe problem for harmonic foliations

The fractional Yamabe problem, proposed by Gonz\'{a}lez-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the…

In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy…

We show that in an arbitrarily fixed conformal class on a closed manifold, the upper bound condition of the total scalar curvature is $C^{0}$-closed if its Yamabe constant is nonpositive. Moreover, we show that if a conformal class on a…

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive?…

Let $ (M, g) $ be a compact manifold or a complete non-compact manifold without boundary, $ \dim M \geqslant 4 $, and not locally conformally flat. In this article, we introduce a new local method to resolve the Yamabe problem on compact…

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a…

In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function,…

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as…

In this paper, we establish the existence of conformal deformations that uniformize fourth order curvature on 4-dimensional Riemannian manifolds with positive conformal invariants. Specifically, we prove that any closed, compact Riemannian…

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate…

After R.~Schoen completed the solution to the Yamabe problem, compact manifolds could be categorized into three classes, depending on whether they admit a metric with positive, non-negative, or only negative scalar curvature. Here we follow…

In this paper, we study scalar curvature rigidity of non-smooth metrics on smooth manifolds with non-positive Yamabe invariant. We prove that if the scalar curvature is not less than the Yamabe invariant in distributional sense, then the…

A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential…

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the…

We first prove stochastic representation formulae for space-time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions.…

Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or…

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than…

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…

In this paper, we consider the Dirichlet boundary value problem for fully nonlinear Yamabe equations on Riemannian manifolds with boundary. Assuming the existence of a subsolution, we derive \emph{a priori} boundary second derivative…