arXiv research

Researchers define residue families and use them to solve singular Yamabe problems.

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

New operators and curvatures derived from embedded manifolds.

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

Derives GJMS operators and Q-curvatures for submanifolds.

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an ext…

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existen…

Study bounds Neumann and Steklov eigenvalues on manifolds and submanifolds.

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

Study of tractor bundles and spacelike immersions in Lorentzian manifolds.

We use the conformal invariance and the holographic correspondence to fully specify the dependence of entanglement entropy on the extrinsic geometry of the 2d surface $Σ$ that separates two subsystems of quantum strongly coupled ${\mathcal{N}}=4$ SU(N) superconformal gauge theory. We extend this result and calculate en…

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

Paper studies smoothness of bi-conformal heat flow on 4-manifolds.

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second…

Analyzes conformal anomaly in five dimensions, identifying new boundary conformal invariants.

Develops methods for computing conformal invariants of submanifolds.

We proved that a conformal immersion of $M_0^{n_0}\times M_1^{n_1}$ as an hipersurface in a Euclidean space must be an extrinsic product of immersions, under the assumption that $n_0, n_1 \geq 2$ and that $M^{n_0}_0\times M^{n_1}_1$ is not conformally flat. We also stated a similar theorem for an arbitrary number of fa…

Paper studies critical points of curvature energies in 4D.

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

The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.

Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.

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.

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of $\bar M$ bounded from below, and obtain an extrinsic…

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

The paper finds universal inequalities for eigenvalues on hyperbolic spaces.

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

Upper bound for Laplacian eigenvalue via conformal volume.

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

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …

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

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …

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…

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…

The Gauss formula is extended to various Laplacians on submanifolds.

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…

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…

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…

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

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…

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…

The paper studies eigenvalues of the Dirac operator on Riemannian manifolds.

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in $H^3$ or in…

Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.

We study the Laplacian flow of a $\mathrm{G}_2$-structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. The…