Conformal geometry and fully nonlinear equations

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

Some properties of the Schouten tensor and applications to conformal geometry

The note is about some nonlinear curvature conditions which arise naturally in conformal geometry.

Invariant Prolongation of BGG-Operators in Conformal Geometry

BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor connection which yields an invariant prolongation of the first operator. The explicit calculation is presented in the case of conformal Killing forms.

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe

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 Liouville type problem.

Recent progress on the fractional Laplacian 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.

Local gauge conditions for ellipticity in conformal geometry

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.

Conformal Geometry on Four Manifolds

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was heavily influenced by many earlier pioneer works; recent progress in conformal geometry has also been made in many different directions, here we will only present some slices of the development.

Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Schouten tensor equations in conformal geometry with prescribed boundary metric

On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.

Local estimates for elliptic equations arising in conformal geometry

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

Invariant prolongation of overdetermined PDE's in projective, conformal and Grassmannian geometry

This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation covariant derivatives for a number of interesting examples in projective, conformal and Grassmannian geometries.

An integral equation in conformal geometry

We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced to an extremal problem for the harmonic extension of functions and the related nonlinear integral equation is discussed.

A class of fully nonlinear equations arising in conformal geometry

In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $σ_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for solutions to these equations and establish an existence result associated to them.

An equation of Monge-Ampere type in conformal geometry, and four-manifolds of positive Ricci curvature

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.

On pseudo-harmonic maps in conformal geometry

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. For example, we show that no co-compact lattice in SO(1,n), n>2, can be the fundamental group of a compact Kahler-Weyl manifold of certain type.

Towards a Liouville theorem for continuous viscosity solutions to fully nonlinear elliptic equations in conformal geometry

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.

Convexity and singularities of curvature equations in conformal geometry

We define a generalization of convex functions, which we call $δ$-convex functions, and show they must satisfy interior Hölder and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative Ricci curvature. We prove that such solutions either extend to a Hölder continuous function across the singularity, or else have the same singular behavior as the fundamental solution of the conformal Laplacian. We also obtain various removable singularity theorems for these equa…

Uniqueness Theorems for fully nonlinear conformal equations on subdomains of the sphere

In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.

Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.

Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants

This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal invariants associated with conformal diffeomorphisms. This includes conformal invariants associated with equivariant characteristic classes. The approach of this paper involves using various tools from noncommutative geometry, such as twisted spectral triples and cyclic theory. …

A fully nonlinear conformal flow on locally conformally flat manifolds

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.

The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.

几何与调和分析中的共形对称性

Conformal symmetries in geometry and harmonic analysis

本文探讨了共形对称性在几何与调和分析中的应用，重点介绍了 Yamabe 运算符及其在相关领域的使用。

CR geometry and conformal foliations

We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.

Jet isomorphism for conformal geometry

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 is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi.

Conformal geometry of non-reductive four-dimensional homogeneous spaces

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

Geodesics and Submanifold Structures in Conformal Geometry

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 the theory of extrinsic conformal geometry for submanifolds, find tensorial invariants of a conformal embedding, and use these invariants to characterize various forms of geodesic submanifolds.

On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds.

New Conformal Invariants in Absolute Parallelism Geometry

The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.

A class of variational functionals in conformal geometry

We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of the metric on the manifold.

Volume growth and puncture repair in conformal geometry

Suppose $M$ is a compact Riemannian manifold and $p\in M$ an arbitrary point. We employ estimates on the volume growth around $p$ to prove that the only conformal compactification of $M\setminus\{p\}$ is $M$ itself.

Conformal Gauss Map Geometry and Application to Willmore Surfaces in Model Spaces

In this paper we make a detailed and self-contained study of the conformalGauss map. Then, starting from the seminal work of R. Bryant and the notion of conformal Gauss map, we recover many fundamental properties of Willmore surfaces. We also get new results like some characterizations of minimal and constant meancurvature (CMC) surfaces in term of their conformal Gauss map behavior.

Conformal submanifold geometry I-III

In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizing immersed submanifolds of the conformal n-sphere. These methods are applied in Part III to study constrained Willmore surfaces, isothermic surfaces, Guichard surfaces and conformally-flat submanifolds with flat normal bundle, and their spectral deformations, in arbitrary codime…

The geometry of conformally Einstein metrics with degenerate Weyl tensor

The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations arising on conformally Einstein spaces (with Riemannian signature) where this condition fails, which then allows us to characterize a general class of locally conformally Einstein Riemannian manifolds with degenerate Weyl tensor.

Non-linear partial differential equations in conformal geometry

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, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced. The conformally covariant powers of the Laplacian form a family $P_{2k}$ with $k \in \mathbb N$ and $k \leq \frac{n}{2}$ if the dimension $n$ is even. Each $P_{2k}$ has leading order term $(- Δ)^k$ and is equal to $ (- Δ) ^k$ if the metric is fl…

Asymptotic geometry and conformal types of Carnot--Carathéodory spaces

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further …

Geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds

In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.

Conformal geometry of the (discrete) Schwarzian Davey-Stewartson II hierarchy

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.

Conformal geometry and special holonomy

A theorem of Lawson and Simons states that the only stable minimal submanifolds in complex projective spaces are complex submanifolds. We generalize their result to the cases of quaternionic and octonionic projective spaces. Our approach gives a unified viewpoint towards conformal and projective geometries.

Noncommutative residue, conformal invariants and lower dimensional volumes in Riemannian geometry

The results of this paper are outdated. Finer versions of them will appear elsewhere.

Noncommutative Geometry and Conformal Geometry. III. Vafa-Witten Inequality and Poincaré Duality

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under co…

Conformal slant submersions in contact geometry

Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $ξ$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions.…

Uniform K-stability and Conformally Kähler, Einstein-Maxwell geometry on toric manifolds

Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.

Errata and Comments for "Energy of knots and conformal geometry"

This article serves as errata of the book "Energy of knots and conformal geometry", Series on Knots and Everything Vol. 33, World Scientific, Singapore, 304 pages, (2003). (ver. 27/05/2007)

紧致辛流形上的共形不变狄拉克型方程：几何的影响

A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry

本文研究了紧致辛流形上的共形不变狄拉克型方程，证明了在非球面情况下Aubin型不等式严格成立，并为四维共形狄拉克-爱因斯坦问题提供了地面态存在性结果。

Conformal geometry of timelike curves in the (1+2)-Einstein universe

We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of $\mathbb{R}^{2,3}$ by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of timelike curves and to address the question of existence and properties of closed trajectories for the conformal strain functional. Some relations between the conformal geometry of timelike curves and the geometry of knots and links in the 3-sphere are discussed.

Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms and twistor spinors. Finally we discuss a class of generic twistor spinors …

On the geometry of conformal geodesics equations

We answer to the question whether a system of the 3rd order ODEs describes geodesics of a conformal structure. We construct a functor from a category of conformal geometries to a category of Cartan geometries associated to the 3rd order ODEs systems. Explicit formulas which define the family of all equations on conformal geodesics are given in the last section of the article.

Jet Finslerian geometry of the conformal Minkowski metric

The paper develops the Finsler-like geometry on the 1-jet space for the jet conformal Minkowski (JCM) metric, which naturally extends the Minkowski metric in the Chernov-Pavlov framework. To this aim there are determined the nonlinear connection, distinguished (d-) Cartan linear connection, d-torsions and d-curvatures. The field geometrical gravitational and electromagnetic d-models based on the JCM metric are discussed.

Conformal geometry of isotropic curves in the complex quadric

Let $\mathbb{Q}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in $\mathbb{Q}_3$. By an isotropic curve we mean a nonconstant holomorphic map from a Riemann surface into $\mathbb{Q}_3$, null with respect to the conformal structure of $\mathbb{Q}_3$. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.

Refined Kato inequalities and conformal weights in Riemannian geometry

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.

曲线上的射影结构与共形几何

Projective structures on curves and conformal geometry

本文梳理曲线上射影结构各等价定义与同构类的对应，纠正文献疏漏，并据此证明共形/Möbius 环境空间中的曲线 Yamabe 问题一般无解。