Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.
problem Solving CR Poincaré-Lelong equation on CR manifolds.
method Solves CR Poisson equation on CR (2n+1)-manifolds with specific curvature properties. result Discovers structures and CR Yamabe steady solitons on complete noncompact Sasakian manifolds.
In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1,1)-forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides …
The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.
problem Analyzing coherent sheaves on complex manifolds using global analytic methods.
method Developing residue currents for cohesive modules and proving their properties.
result Proves a generalized Poincaré-Lelong formula for cohesive modules.
The paper studies Fubini-Study metrics and zero distributions on CR manifolds.
problem Understanding Fubini-Study metrics on CR manifolds.
method Asymptotic analysis of Toeplitz operators and pull-back metrics.
result Established the distribution of zero divisors of random CR functions.
We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss …
Generalizes double transgression formulas on complex manifolds.
problem Currential double transgression formulas on complex manifolds.
method General framework using Bott-Chern Duality.
result Complements existing transgression formulas and provides new applications.
We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the pap…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…
Paper proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
problem Proving Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
method Analyzes CR-manifolds with contact structure conformal to Heisenberg group, proving volume form is a strong A_infinity weight.
result Proves Sobolev-Poincaré inequality for CR-manifolds with integrable Q-curvature.
The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.
problem Embedding CR manifolds into twistor spaces and constructing neutral hyperkähler metrics.
method Embedding a real analytic twistor CR manifold into the twistor space of a Poincaré-Einstein metric, constructing the associated Fefferman ambient metric as a neutral hyperkähler metric.
result The construction of neutral hyperkähler metrics associated with twistor CR manifolds.
Kähler-Ricci flow smooths out positive closed currents with divisorial singularities
problem Geometric regularization of positive closed currents
method Kähler-Ricci flow
result Gradual replacement of divisorial singularities by Poincaré type ones
This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps α:E→F which, for smooth connections on E and F, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…
The CR Yamabe flow converges exponentially to a contact form with flat curvature.
problem Analyzing the CR Yamabe flow with zero invariant.
method Used CR Poincaré inequality and Gagliardo-Nirenberg type interpolation inequality.
result The flow converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature.
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
Solves a complex Monge-Ampère equation on compact Hermitian manifolds.
problem Solving a specific Monge-Ampère equation on compact Hermitian manifolds.
method Uses complex Monge-Ampère equation and fixed potential approach.
result Shows the existence and uniqueness of a solution in a specific class.
We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of…
Survey on Einstein metrics on domains, linking complex geometry to CR structures.
problem Existence of asymptotically complex hyperbolic Einstein metrics.
method Bulk-boundary correspondence between CR structures and Einstein metrics.
result Existence theorems for Einstein metrics on strictly pseudoconvex domains.
Extends Lelong number theory to positive plurisubharmonic currents.
problem Lack of in-depth exploration of Lelong number theory for positive plurisubharmonic currents.
method Introduces generalized Lelong numbers and studies their properties using Lelong-Jensen formulas for the normal bundle.
result Shows the top degree Lelong number of a positive plurisubharmonic current is totally intrinsic.
The study finds solutions for CR spheres with a curvature condition.
problem Prescribing scalar curvature on Cauchy-Riemann spheres.
method Bahri methods and critical points at infinity theory.
result Lower bound for the number of solutions found.
The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.
problem Defining and studying generalized Lelong numbers for currents in intersection theory.
method Formulating generalized Lelong numbers for closed smooth (j,j)-forms, defining horizontal dimension, and establishing properties and formulas.
result Effective sufficient conditions for defining and continuity of intersections of positive closed currents.
The paper studies symmetrization effects on Lelong numbers and Monge-Ampère masses of plurisubharmonic functions.
problem Analyzing symmetrization effects on Lelong numbers and Monge-Ampère masses of plurisubharmonic functions.
method Schwarz symmetrization of S1-invariant plurisubharmonic functions on balanced domains in Cn. result The Monge-Ampère mass decreases under Schwarz symmetrization for toric functions with a single pole at the origin.
We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dime…
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.
Positive mass theorem and Yamabe equation on CR manifolds
problem Positive mass theorem and Yamabe equation on CR manifolds
method Positive mass theorem and Yamabe equation on CR manifolds
result Positive mass theorem in 3-dimensional CR geometry
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Analyzing the residual Monge-Ampère mass of symmetric plurisubharmonic functions.
method Proved zero mass for functions with zero Lelong number at origin and S1-invariance. result Zero mass conjecture answered for symmetric functions.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
Two types of nonvanishing results are presented for compact Kähler varieties.
problem Deriving geometric consequences from numerical information in Kähler geometry.
method Analyzing non-uniruled varieties and hyperkähler manifolds to establish nonvanishing results for adjoint and nef bundles.
result Strong abundance-type results are obtained in dimension 4.
Holomorphic curves exiting bounded symmetric domains are asymptotically totally geodesic.
problem Understanding the asymptotic behavior of holomorphic curves in bounded symmetric domains.
method Proof by contradiction and rescaling, using the Poincaré-Lelong equation.
result Holomorphic curves exiting a bounded symmetric domain are asymptotically totally geodesic.
Researchers study surface area functionals in CR manifolds, deducing equations for various cases.
problem Investigating surface area functionals in 3D CR manifolds.
method Deduced Euler-Lagrange equations for energy functionals in various 3D CR manifolds.
result New equations deduced for surface area functionals on disk bundles, Rossi spheres, and 3D tori.
Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
Paper studies fractional CR Yamabe equation on sphere, proving multiplicity of solutions.
problem Fractional CR Yamabe equation on sphere solutions.
method Analyzed Palais-Smale sequences to characterize bubbling phenomena and prove multiplicity of solutions.
result Existence of infinitely many solutions to the fractional CR Yamabe equation.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Estimating the residual Monge-Ampère mass of symmetric plurisubharmonic functions with isolated singularities.
method Utilized Sasakian geometry to derive estimates on the residual mass in relation to Lelong numbers.
result Partially resolved the zero mass conjecture by Guedj and Rashkovskii.
In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…
CR 3-sphere rigidity proven through curvature invariant.
problem Proving rigidity of CR 3-sphere under deformations.
method Analyzing curvature invariant and linearized equation.
result CR 3-sphere does not admit nontrivial obstruction flat deformations.
Derives a simpler equation for CR-curvature vanishing on certain complex hypersurfaces.
problem Deriving a simpler equation for CR-curvature vanishing on certain complex hypersurfaces.
method Utilizes two invariants discovered by S. Pocchiola to provide an alternative derivation of the CR-curvature vanishing condition.
result Provides an alternative derivation of the CR-curvature vanishing condition equivalent to the Monge equation.
We discuss relations between the para-CR structures and differential equations (both ODEs and PDEs of finite type).
A new direct construction method for Cartan-Moser chains.
problem Detecting Cartan-Moser chains from advanced considerations.
method Inspection of Lie prolongations of infinitesimal automorphisms.
result Found a simple cubic degenerate orbit locus.
We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discus…
Reduces a complex hypersurface to a simplified equation with primary invariants.
problem Analyzing Levi degenerate CR manifolds in 5 dimensions.
method Applying Lie's theory, integrating and straightening chains, and using Poincaré-Moser reduction.
result Shows a convergent change of coordinates that simplifies the equation of the manifold.
Estimates Lelong numbers of Monge-Ampère products for Kähler manifolds.
problem Estimating Lelong numbers of Monge-Ampère products on Kähler manifolds.
method Analyzes generalized Monge-Ampère products and applies estimates to Chern and Segre currents of pseudoeffective vector bundles.
result Generalizes a recent result about pseudoeffective vector bundles and their nefness.
CR-harmonic maps defined for pseudoconvex manifolds.
problem Defining CR-harmonic maps in CR geometry.
method Developing renormalized energy and CR covariant subelliptic PDE.
result CR-harmonic maps satisfy a CR covariant subelliptic PDE.
New Liouville-type results for CR Yamabe equation in Heisenberg group.
problem Characterizing solutions to CR Yamabe equation in Heisenberg group.
method Integral estimates combined with divergence formula.
result Liouville-type results for bounded solutions in n=2 and solutions with pointwise decay assumption in n≥3. Sharp inequalities on Siegel domains and complex hyperbolic spaces established.
problem Establishing inequalities on complex hyperbolic spaces and Siegel domains.
method Helgason-Fourier analysis, Kunze-Stein phenomenon, factorization theorem.
result Sharp Hardy-Adams and Adams type inequalities on Sobolev spaces of any positive fractional order on complex hyperbolic spaces.
Study shows nonvanishing CR curvature on Grauert tube boundaries.
problem Analyzing CR curvature on Grauert tube boundaries.
method Two recent formulas for Cartan CR-curvature of local smooth hypersurfaces in C^2.
result Nonvanishing Cartan CR-curvature on Grauert tube boundaries.
Study minimizes CR surfaces in Heisenberg group with rotational symmetry.
problem Minimizing CR surfaces with vanishing CR invariant energy E1 in Heisenberg group. method Proved local uniqueness, classified global surfaces with rotational symmetry, computed second variation.
result Clifford torus is not a local minimizer of E1. Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.
problem Characterizing and simplifying the defining equations of 2-nondegenerate CR hypersurfaces.
method Characterization of 2-nondegenerate models, derivation of normal forms, computation of CR invariants, derivation of infinitesimal symmetries.
result The moduli space of 2-nondegenerate CR hypersurfaces in C^N is infinite dimensional for N>3.
Study on cr-invariant variational problem for Legendrian curves in 3-sphere.
problem Lower-order cr-invariant variational problem for Legendrian curves in 3-sphere.
method Deduced Euler-Lagrange equations, investigated closed critical curves, characterized non-constant cr-curvature curves, proved cr-equivalence classes correspondence to rational points.
result Closed critical curves with non-constant cr-curvature are characterized and their cr-equivalence classes are in one-to-one correspondence with rational points of a connected planar domain.