arXiv research

Proves existence and uniqueness of weighted metrics for smooth spaces.

We prove existence and uniqueness of weighted ambient metric for manifolds with density.

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

The conformal Fefferman-Graham ambient metric construction is one of the most fundamental constructions in conformal geometry. It embeds a manifold with a conformal structure into a pseudo-Riemannian manifold whose Ricci tensor vanishes up to a certain order along the original manifold. Despite the general existence re…

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal …

The paper proves geometric inequalities for hypersurfaces in weighted manifolds.

The paper characterizes ambient metrics using conformal completion and null infinity properties.

An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedur…

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which th…

In this paper we relate the Fefferman-Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tract…

Extends Weyl geometry from conformal to Weyl manifolds using ambient metrics.

This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are considered. Equivalence with the expansions of associated Poincare metrics is e…

This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As…

We present conformal structures in signature (3,2) for which the holonomy of the Fefferman-Graham ambient metric is equal to the non-compact exceptional Lie group G_{2(2)}. We write down the resulting 8-parameter family of G_{2(2)}-metrics in dimension seven explicitly in an appropriately chosen coordinate system on th…

Novel model detects communities in noisy multilayer networks.

This paper studies the relation between two notions of holonomy on a conformal manifold. The first is the conformal holonomy, defined to be the holonomy of the normal tractor connection. The second is the holonomy of the Fefferman-Graham ambient metric of the conformal manifold. It is shown that the infinitesimal confo…

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conforma…

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally …

In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

We provide an explicit formula for the Fefferman-Graham-ambient metric of an $n$-dimensional conformal $pp$-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of Bach-flat examp…

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as aff…

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR g…

These notes are based on my lectures at IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations." Here I try to explain basic ideas of the ambient metric construction by studying the Szego kernel of the sphere.

Researchers create a family of conformally covariant operators.

Extended solitons show constant curvature on compact manifolds.

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show…

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operat…

New geometric analysis of PWSPDs balances density and geometry in high-dimensional data.

Generative models use Riemannian manifolds to improve latent space interpretation.

In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into $\R^\ell$. We are also lead to study the co…

Generalizes embedding formalism for CFTs on curved backgrounds.

The study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.

In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for the geometry of $(2, 3, 5)$ distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type $\textrm{G}_2$. He claimed to produce a local classification of all such (complex) dis…

We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…

Study of weighted nonlinear flags in symplectic geometry.

Let $Σ$ be a compact immersed surface with constant weighted mean curvature $H_f$ in a weighted manifold $(M^3,g,f)$. In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on $Σ$ in terms of $H_f$ and the curvature of the ambient. As consequence we obtain that there is no stable …

Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension …

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields…

Modified construction for conformal structures with twistor spinors.

The paper proves rigidity for shells in non-Euclidean spaces.

Study mean curvature flow into evolving manifold with coupled flows.

We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the $k^{\mathrm{th}}$ mean curvature in terms of the area of the hypersurf…

Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geom…

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--S…

Paper proves existence of ambient manifolds for null hypersurfaces solving Einstein equations.

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…