Related papers: On some conformally invariant fully nonlinear equa…

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally…

Let $X$ be an asymptotically hyperbolic manifold and $M$ its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on $M$ under various geometric assumptions on $X$ and $M$: Firstly,…

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes…

Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various…

We define a new formal Riemannian metric on a conformal class in the context of the $v_{\frac{n}{2}}$-Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover,…

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions…

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 establish theorems on the existence and compactness of solutions to the $\sigma_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear…

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe…

We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar…

We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta, we compute these invariants in many cases that were…

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…

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact)…

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain…

In the present paper we prove Liouville-type theorems: non-existence theorems for conformal mappings of complete Riemannian manifolds. In addition, we give an application of these results to the theory of conharmonic transformations. A part…

We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly…

We study solutions and supersolutions of homogeneous and nonhomogeneous $\mathcal{A}$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is…

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation…

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…

We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate…