arXiv subjects

Taiji Marugame

Publications and source records attributed to Taiji Marugame.

13 recordsLinked to original sources

arXiv2026

Ambient and Poincar\'e metrics for CR $3$-manifolds associated with the self-dual Einstein ACH metric

We construct new ambient and Poincar\'e metrics for the Fefferman conformal manifold over $3$-dimensional CR manifolds, starting from the associated self-dual Einstein ACH metric. Unlike the classical ambient and Poincar\'e metrics arising from approximate solutions to the complex Monge-Amp\`ere equation, these metrics satisfy Einstein-Maxwell-type equations to infinite order rather than the Ricci-flat or Einstein equations. Since these metrics are determined to infinite order without ambiguity, they enable us to construct CR invariant differential operators and local CR invariants involving arbitrarily high order derivatives of the Tanaka-Webster curvature and torsion. As an application, we obtain the ambient metric construction of the CR GJMS (Gover-Graham) operators of all orders in dimension $3$.

math.DG↗

arXiv2024

The Fefferman Metric for Twistor CR Manifolds and Conformal Geodesics in Dimension Three

We give an explicit description of the Fefferman metric for twistor CR manifolds in terms of Riemannian structures on the base conformal 3-manifolds. As an application, we prove that chains and null chains on twistor CR manifolds project to conformal geodesics, and that any conformal geodesic has lifts both to a chain and a null chain. By using this correspondence, we give a variational characterization of conformal geodesics in dimension three which involves the total torsion functional.

math.DG↗

arXiv2023

Hyperk\"ahler ambient metrics associated with twistor CR manifolds

Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as $\mathbb{P}^1$-bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space of the anti self-dual Poincar\'e-Einstein metric whose conformal infinity is the base conformal 3-manifold, and construct the associated Fefferman ambient metric as a neutral hyperk\"ahler metric on the spinor bundle with the zero section removed. We also describe the structure of the Cheng--Yau type K\"ahler-Einstein metric which has the twistor CR manifold as the boundary at infinity.

math.DG↗

arXiv2021

The Bonnet theorem for statistical manifolds

We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss--Codazzi--Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian manifold). The proof is based on the notion of statistical embedding to the product of a vector space and its dual space introduced by Lauritzen. As another application of Lauritzen's embedding, we show that a statistical manifold admitting an affine embedding of codimension 1 or 2 is locally embeddable to a flat statistical manifold of the same codimension.

math.DG↗

arXiv2020

Renormalized characteristic forms of the Cheng--Yau metric and global CR invariants

For each invariant polynomial $Φ$, we construct a global CR invariant via the renormalized characteristic form of the Cheng--Yau metric on a strictly pseudoconvex domain. When the degree of $Φ$ is 0, the invariant agrees with the total $Q'$-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant $\mathcal{I}'_Φ$ which integrates to the corresponding CR invariant. These are generalizations of the $\mathcal{I}'$-curvature on CR five-manifolds, introduced by Case--Gover.

math.DG↗

arXiv2020

Global secondary CR invariants in dimension five

A global secondary CR invariant is defined as the integral of a pseudo-hermitian invariant which is independent of a choice of pseudo-Einstein contact form. We prove that any global secondary CR invariant on CR five-manifolds is a linear combination of the total $Q'$-curvature, the total $\mathcal{I}'$-curvature, and the integral of a local CR invariant.

math.DG↗

arXiv2019

Self-dual Einstein ACH metrics and CR GJMS operators in dimension three

By refining Matsumoto's construction of Einstein ACH metrics, we construct a one parameter family of ACH metrics which solve the Einstein equation to infinite order and have a given three dimensional CR structure at infinity. When the parameter is 0, the metric is self-dual to infinite order. As an application, we give another proof of the fact that three dimensional CR manifolds admit CR invariant powers of the sublaplacian (CR GJMS operators) of all orders, which has been proved by Gover-Graham. We also prove the convergence of the formal solutions when the CR structure is real analytic.

math.DG↗

arXiv2019

Chains in CR geometry as geodesics of a Kropina metric

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspondence between geodesics of the Kropina metric and chains allows us to use the methods of metric geometry and the calculus of variations to study chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985] that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain. Finally, we generalize this result to the global setting by showing that any two points of a connected compact strictly pseudoconvex CR manifold which admits a pseudo-Einstein contact form with positive Tanaka-Webster scalar curvature can be joined by a chain.

math.DG↗

arXiv2018

Some Remarks on the Total CR $Q$ and $Q^\prime$-Curvatures

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

math.DG↗

arXiv2017

Volume renormalization for the Blaschke metric on strictly convex domains

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate an intrinsic geometry of the boundary as a conformal Codazzi structure and show that L gives a global conformal invariant of the boundary.

math.DG↗

arXiv2016

Variation of total Q-prime curvature on CR manifolds

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gover and the first author, we study the property of CR invariant differential operator of order 2(n+3) on 2n+1 dimensional CR manifolds that appears in the second variational formula of the total Q-prime curvature.

math.DG↗

arXiv2016

Renormalized Chern-Gauss-Bonnet formula for complete Kahler-Einstein metrics

We present a renormalized Gauss-Bonnet formula for approximate Kahler-Einstein metrics on compact complex manifolds with pseudo-Einstein CR boundaries. The boundary integral is given explicitly, and it is proved that it gives a pseudo-Einstein invariant, which generalizes the Burns-Epstein invariant.

math.CV↗

arXiv2016

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.

math.DG↗