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Yuya Takeuchi

Publications and source records attributed to Yuya Takeuchi.

At least 19 recordsLinked to original sources
arXiv2026

Joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds

A Heisenberg Bieberbach manifold is a compact quotient of the Heisenberg group by a discrete torsion-free subgroup of its pseudo-Hermitian automorphism group. In this paper, we study the joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds. In particular, we explicitly compute the multiplicities of the joint spectrum using theta functions and the character theory of finite groups.

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arXiv2026

Non-embeddable torus and CR Paneitz operator

The CR Paneitz operator is closely related to several important problems in CR geometry. In this paper, we study the CR Paneitz operator on non-embeddable three-dimensional tori. Under mild assumptions, we show that it possesses infinitely many negative eigenvalues. We also provide concrete examples satisfying the assumptions.

math.DG↗
arXiv2026

Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

We study the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature on non-compact strictly pseudoconvex CR manifolds. We prove that, under mild assumptions, the universal cover of a compact strictly pseudoconvex CR manifold admits infinitely many non-homothetic such contact forms whenever its fundamental group has infinite profinite completion. As applications, we treat complements of real or complex spheres in the standard CR sphere, as well as circle bundles over compact K\"{a}hler manifolds and the boundary of a Reinhardt domain.

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arXiv2025

CR Paneitz operator and embeddability

In this article, we give a brief survey of recent developments on relations between global embeddability of a closed strictly pseudoconvex CR manifold and the CR Paneitz operator.

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arXiv2024

CR Paneitz operator on non-embeddable CR manifolds

The CR Paneitz operator is closely related to some important problems in CR geometry. In this paper, we consider this operator on a non-embeddable CR manifold. This operator is essentially self-adjoint and its spectrum is discrete except zero. Moreover, the eigenspace corresponding to each non-zero eigenvalue is a finite dimensional subspace of the space of smooth functions. Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an example of non-embeddable CR manifolds, has infinitely many negative eigenvalues, which is significantly different from the embeddable case.

math.CV↗
arXiv2023

Kohn-Rossi cohomology of spherical CR manifolds

We prove some vanishing theorems for the Kohn-Rossi cohomology of some spherical CR manifolds. To this end, we use a canonical contact form defined via the Patterson-Sullivan measure and Weitzenb\"{o}ck-type formulae for the Kohn Laplacian. We also see that our results are optimal in some cases.

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arXiv2022

CR Yamabe constant and inequivalent CR structures

The CR Yamabe constant is an invariant of a compact strongly pseudoconvex CR manifold and plays an important role in CR geometry. We show some integral formulae of the CR Yamabe constant. We also construct an infinite-dimensional family of strongly pseudoconvex CR structures with varying CR Yamabe constants and a compact simply-connected manifold admitting two strongly pseudoconvex CR structures with different signs of the CR Yamabe constant.

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arXiv2022

Generalizations of the $Q$-prime curvature via renormalized characteristic forms

The $Q$-prime curvature is a local pseudo-Einstein invariant on CR manifolds defined by Case and Yang, and Hirachi. Its integral, the total $Q$-prime curvature, gives a non-trivial global CR invariant. On the other hand, Marugame has constructed a family of global CR invariants via renormalized characteristic forms, which contains the total $Q$-prime curvature. In this paper, we introduce a generalization of the $Q$-prime curvature for each renormalized characteristic form, and show that its integral coincides with Marugame's CR invariant. We also study generalizations of the critical CR GJMS operator and the $P$-prime operator, which are related to the transformation laws of our new curvatures under conformal change.

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arXiv2020

Analysis of the critical CR GJMS operator

The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.

math.DG↗
arXiv2020

Chern classes of spherical CR manifolds

We first construct closed spherical CR manifolds of dimension at least five having non-trivial first Chern class with real coefficients. We next prove a constraint on Chern classes with real coefficients of (not necessarily closed) spherical CR manifolds. Finally, we obtain a topological obstruction to the existence of spherical CR structures on co-oriented contact manifolds.

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arXiv2020

$\mathcal{I}^\prime$-curvatures in higher dimensions and the Hirachi conjecture

We construct higher-dimensional analogues of the $\mathcal{I}^\prime$-curvature of Case and Gover in all CR dimensions $n\geq2$. Our $\mathcal{I}^\prime$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n\geq2$.

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arXiv2019

Nonnegativity of the CR Paneitz operator for embeddable CR manifolds

The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a contact form with zero CR $Q$-curvature, and generalize the total $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einstein contact forms. Furthermore, we discuss the logarithmic singularity of the Szeg\H{o} kernel.

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arXiv2018

Stability of the existence of a pseudo-Einstein contact form

A pseudo-Einstein contact form plays a crucial role in defining some global invariants of closed strictly pseudoconvex CR manifolds. In this paper, we prove that the existence of a pseudo-Einstein contact form is preserved under deformations as a real hypersurface in a fixed complex manifold of complex dimension at least three.

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arXiv2018

On the second variation of the Graham-Witten energy

The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal submanifolds in an Einstein manifold. As an application, we prove that the even-dimensional totally geodesic spheres in the unit sphere are critical points of the Graham-Witten energy with non-negative second variation.

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arXiv2017

Ambient constructions for Sasakian $\eta$-Einstein manifolds

The theory of ambient spaces is useful to define CR invariant objects, such as CR invariant powers of the sub-Laplacian, the $P$-prime operators, and $Q$-prime curvature. However in general, it is difficult to write down these objects in terms of the Tanaka-Webster connection. In this paper, we give those explicit formulas for CR manifolds satisfying an Einstein condition, called Sasakian $\eta$-Einstein manifolds. As an application, we study properties of the first and the second variation of the total $Q$-prime curvature at Sasakian $\eta$-Einstein manifolds.

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