arXiv research

The fractional Yamabe problem, proposed by González-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 classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…

The paper studies inequalities for fractional GJMS operators on conformal infinity.

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 geometric conditions on $X$ and $M$. We also obtain the existence results for t…

Study finds non-uniqueness in sphere metrics with constant fractional curvature.

Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

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 boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincaré-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classic…

We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-Δ)^γw = c_{n, γ} w^{\frac{n+2γ}{n-2γ}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…

Study CR Yamabe constant and CR structures on manifolds.

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely…

Study CR Yamabe constant, flow, and soliton on CR manifolds.

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^γ[g^+,\hat{h}] (u) = cu^{n+2γ\over n-2γ}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^γ[g^+,\hat{h}]$ is the fractiona…

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…

The study compares and finds Yamabe constants on warped products.

Fractional combinatorial flow improves surface conformal structures.

Paper shows k-Yamabe solitons have constant curvature under certain conditions.

Study convergence of Yamabe flow on singular spaces with positive constant.

We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.

The Yamabe flow converges to a specific function on compactified manifolds.

Study shows long-term flow on special manifolds with positive Yamabe constant.

The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.

Paper finds conditions for non-Einstein relative Yamabe metrics.

Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is …

Study on warped product Yamabe solitons with constant fiber curvature.

Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and (W,h).

Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.

Suppose $M_{1}$ and $M_{2}$ are $3$-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a CR structure with positive CR Yamabe constant.

Assume that $(X, g^+)$ is an asymptotically hyperbolic manifold, $(M, [\bar{h}])$ is its conformal infinity, $ρ$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g} = ρ^2 g^+$. For any $γ\in (0,1)$, we prove that the solution set of the $γ$-Yamabe problem on $M$ is compact in $C^2(M)$ provid…

Suppose $M_{1}$ and $M_{2}$ are two closed (compact with no boundary) spherical CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $M_{1}$ and $M_{2}$ also admits a spherical CR structure with positive CR Yamabe constant.

The paper classifies expanding gradient Yamabe solitons based on scalar curvature.

Paper shows constant σk-curvature for quasi k-Yamabe solitons.

Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.

In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.

The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.

Proves product metrics are Yamabe metrics under small flat torus conditions.

New approach linking CR Yamabe invariant to Sasaki structures.

The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.

Let $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times N,g+th)$ as $t$ goes to $+\infty$. We obtain that $\lim_{t \to +\infty}Y^2(M\times N,[…

The study classifies specific types of solitons with bounded scalar curvature.

The paper proves solutions for Yamabe equations on manifolds with boundary.

Characterizes gradient Yamabe solitons with specific conditions.

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 notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.