The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.
problem Proving the Lorentzian conformal Lichnerowicz conjecture in locally homogeneous settings.
method Analyzing conformal groups on plane waves and proving the conjecture in a specific setting.
result The Lorentzian conformal Lichnerowicz conjecture is proven in a locally homogeneous setting.
New order defined for conformal classes, impacts Bartnik's conjecture.
problem Bartnik's conjecture and its implications under different energy conditions.
method Defined a new order on conformal classes and analyzed implications under null energy condition.
result The null energy condition can lead to future complete metrics in any dimension.
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.
We extend Garsia's conjecture about surface embeddings.
problem Realizing conformal classes of genus-1 surfaces via embeddings.
method Formulated within the framework of connections on principal bundles, solutions provided in several cases.
result Novel parameterizations of the moduli space of conformal classes of compact surfaces of genus 1.
Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a…
A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetr…
Proves a conjecture about metrics and minimal area enclosures.
problem Proving a conjecture about metrics and minimal area enclosures.
method Using boundedness of harmonic function u, proving the conjecture for asymptotically flat 3-manifolds.
result Proves the bounded conformal conjecture under the assumption of boundedness of harmonic function u.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as…
This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Solves a conjecture using a new formula on conformally Einstein manifolds.
problem Solving a conjecture in conformal geometry.
method Using an Obata type formula established by previous works.
result Solves Hang-Yang conjecture via an Obata-type argument.
Study on Yamabe solitons with applications and structure elucidation.
problem Understanding the structure of Yamabe solitons and their applications.
method Investigation of complete gradient conformal solitons under specific conditions.
result Affirmative partial answer to Yamabe soliton conjecture.
The abstract discusses conjectures about metrics on complex manifolds.
problem The abstract tackles the conjectures about metrics on complex manifolds, specifically balanced, SKT, and LCK.
method The abstract uses complex Hermitian manifolds, closed 1-forms, and conjectures to explore these metrics.
result The abstract verifies a conjecture about the Bott--Chern homology for all known classes of LCK manifolds.
This is the fifth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed a…
This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dime…
Proves Lorentzian manifold properties for analytic 3D spaces.
problem Analyzing properties of Lorentzian manifolds.
method Analyzes compact, real-analytic, three-dimensional Lorentzian manifolds.
result Identity conformal group preserves metric or manifold is flat.
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
We give an overview of the constrained Willmore problem and address some conjectures arising from partial results and numerical experiments. Ramifications of these conjectures would lead to a deeper understanding of the Willmore functional over conformal immersions from compact surfaces.
This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Study explores solvable Lie groups' actions on closed Lorentzian manifolds.
problem Understanding conformal actions of solvable Lie groups on closed Lorentzian manifolds.
method Analyzes the identity component of the conformal group and uses algebraic hypotheses.
result Establishes conditions for conformal flatness and local embeddings.
This note (originally from 2015) provides a proof of a 1985 conjecture of Montiel and Ros concerning the conformal volume of tori. This updated version adds a proof of the claim made in Remark 5 about the value of the conformal volume of tori in the cases not covered by the conjecture of Montiel and Ros. Originally, I …
Paper solves long-standing Gaussian curvature conjecture for minimal graphs.
problem Gaussian curvature of minimal graphs over the unit disk.
method Complex-analytic methods, conformal harmonic parameterization.
result Sharp estimate for Gaussian curvature at the origin of minimal graphs.
Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called B…
We construct the first known examples of compact pseudo-Riemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types (p,q), with 2≤p≤q.
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also stu…
Consider a geometrically finite Kleinian group G without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. Suppose that for each boundary component of M, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, w…
Extends Gromov non-squeezing to locally conformally symplectic structures.
problem Generalizing Gromov non-squeezing to new geometric structures.
method Deformation theory applied to locally conformally symplectic structures.
result Proves a new extension of the Gromov non-squeezing phenomenon.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
The main goal of this paper is to show a counterexample to the following conjecture: {\bf Conjecture} [Meeks, Sullivan]: If f:M→R3 is a complete proper minimal immersion where M is a Riemannian surface without boundary and with finite genus, then M is parabolic. We have proved: {\bf Theorem:} There e…
The study characterizes Riemannian manifolds with conformal vector fields and proves isometric properties.
problem Characterizing Riemannian manifolds with conformal vector fields and boundary conditions.
method Analyzing the properties of conformal vector fields on compact Riemannian manifolds with or without boundary.
result Proves isometric properties of Riemannian manifolds under specific conditions.
The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.
problem Preventing the existence of null geodesic lines in spacetimes.
method Identifying geometric conditions on foliations of spacetimes that prevent null geodesic lines, especially for spacetimes with compact Cauchy hypersurfaces.
result Conditions on foliations can prevent null geodesic lines, leading to restrictions on cosmological spacetime geometry.
We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in \cite{mmp}. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in \cite{p} and \cite{mmp}. We also show,…
Proves properties of 4-manifolds with scalar curvature constraints.
problem Characterizing 4-manifolds with specific scalar curvature properties.
method Analyzes locally conformally flat metrics and uses Schoen's conjecture.
result Affirmatively answers Noronha's question about 4-manifolds with scalar curvature zero.
Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.
Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.
problem Analyzing volume growth and verifying Cohn-Vossen inequality in locally conformally flat manifolds.
method Refined singularity estimate and characterization of volume growth.
result Analytically characterizes volume growth and verifies Cohn-Vossen inequality.
New proof confirms noncompact locally conformally flat manifolds are compact.
problem Rigidity of Schouten tensor under conformal transformations.
method Proof of Cheng's theorem using modified Schouten tensor.
result Noncompact locally conformally flat manifolds are compact.
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
Study of conformal limits in Nakajima quiver varieties.
problem Understanding the conformal limits of Nakajima quiver varieties.
method Defined and studied a conformal limit construction for Nakajima quiver varieties, proving it is a limit of a one-parameter family and gives a biholomorphic map.
result Proved the conformal limit is a biholomorphic map between Lagrangian submanifolds of different quiver varieties.
We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq …
We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy the so-called 5/4-conjecture.
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Let M be a smooth manifold equipped with a conformal structure, E[w] the space of densities with the the conformal weight w and $D_{w,w+\de}$ the space of differential operators from E[w] to E[w+δ]. Conformal quantization Q is a right inverse of the principle symbol map on Dw,w+δ such that Q is confo…