arXiv research

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

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an e…

In this note, we study the connection between the fractional Laplacian operator that appeared in the recent work of Caffarelli-Silvestre and a class of conformally covariant operators in conformal geometry.

The aim of this paper is to report on recent development on the conformal fractional Laplacian, both from the analytic and geometric points of view, but especially towards the PDE community.

Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.

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

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.

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Lap…

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…

We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our fa…

Study optimal partitions on spheres using fractional Q-curvature and variational methods.

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…

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, we handle when the boundary $M$ has a point at which the mean curvature is negat…

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only …

Fractional Laplacian inverse problem solved for connection Laplacians.

Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.

Unified treatment of two extension problems using heat equation in Heisenberg group.

In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our argumen…

Develops nonparametric regression for non-smooth functions using fractional Laplacian.

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

Fractional combinatorial flow improves surface conformal structures.

In this paper, we investigate eigenvalues of Laplacian on a bounded domain in an $n$-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to A. D. Melas [15]. On the other hand, for the case of fractional Laplacian $(-Δ)^{α/2}|_{D}$, wher…

The paper finds the minimum number of negative eigenvalues for conformal Laplacian metrics.

Magnitude study on manifolds using fractional Laplacian.

Researchers study fractional porous medium equation on hyperbolic space.

Study fractional perimeter asymptotics on Riemannian manifolds as $s$ approaches 0.

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

Maps commuting with sub-Laplacians on Carnot groups are conformal.

In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on $R^n$. We show that the problem has infinite positive solutions in $C^τ(R^n)\bigcap H^α_{loc}(R^n)$. Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.

The paper finds universal inequalities for eigenvalues on hyperbolic spaces.

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…

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…

Upper bound for Laplacian eigenvalue via conformal volume.

The study finds a metric that maximizes the second eigenvalue of the Conformal Laplacian.

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescri…

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…

The paper extends Heintze-Karcher inequalities to fractional Q-curvature.

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and c…

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power…

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.

The article recovers tensor fields from partial data using weighted divergent ray transforms.

In this note we give a simple relation between conformal mapping and the first eigenvalue of Laplacian for surfaces in Euclidean spaces.

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

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential ope…