Related papers: Yamabe problems for formally self-adjoint, conform…
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.
We study the conformal logarithmic Laplacian on the sphere, an explicit singular integral operator that arises as the derivative (with respect to the order parameter) of the conformal fractional Laplacian at zero. Our analysis provides a…
We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formul{\ae} for…
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $\sigma_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is…
We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions of families of such operators. In one direction, we show that any homogeneous conformally variational scalar…
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding…
In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role. Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition,…
We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $S^1$ inside $S^m$, $m\geq 5$,…
There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact poses a serious question on the physical meaning of such a…
In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several…
In this essay we give an introduction to conformal symmetry, based on the example of the Yamabe operator and its use in conformal differential geometry, and in representation theory.
In this paper, we introduce the conformal fractional--logarithmic Laplacian on the unit sphere, defined as the derivative of the conformal fractional Laplacian with respect to the order parameter \(s\in(0,1)\). We investigate its…
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…
The symmetric Al-Salam--Chihara polynomials for $q>1$ are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on $\ell^2(\Z)$ to which these polynomials are eigenfunctions. We determine…
In this paper we introduce the conformal fractional Dirac operator and its associated fractional spinorial Yamabe problem. We also present a Caffarelli-Silvestre type extension for this fractional operator, allowing us to express it as a…
In this paper we formulate new curvature functions on $\mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not…
We start by taking the analytical approach to discuss how the minimizer of Yamabe functional provides constant scalar curvature and its relationship with the Sobolev Space $W^{1,2}.$ Then, after demonstrating the importance of the sphere…
Let (M,g) be a compact manifold of dimension n greater or equals to 3. We suppose that g is a given metric in a precised Sobolev space and there is a point P in M and d>o such that g is smooth on the ball B(P,d). We define the second Yamabe…
We discuss a functional model for multi--diagonal selfadjoint operators with almost periodic coefficients that generalizes the well known model for finite band Jacobi matrices. It give us an opportunity to construct examples of almost…
We prove that the curved Ovsienko--Redou operators and a related family of differential operators are formally self-adjoint. This verifies two conjectures of Case, Lin, and Yuan.