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Yoshihiko Matsumoto

Publications and source records attributed to Yoshihiko Matsumoto.

14 recordsLinked to original sources
arXiv2026

CR-invariant energy of Legendrian knots in the Heisenberg group

We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\mathcal{H}$, which serves as a sub-Riemannian analog of the Möbius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Korányi distance on $\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\mathbb{R}$-circles in $\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg analog of the Doyle--Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in $\mathcal{H}\times\mathcal{H}$, providing a partial analog of the infinitesimal cross ratio interpretation known from the classical setting.

math.GT↗
arXiv2025

Asymptotic Dirichlet problem for harmonic maps and conformal geodesics

The asymptotic Dirichlet problem for harmonic maps from the hyperbolic plane into conformally compact Einstein manifolds is used to give a holographic characterization of conformal geodesics on the boundary at infinity, in a way deeply inspired by a work of Fine and Herfray on renormalized area minimization.

math.DG↗
arXiv2022

The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry

We elaborate the tractor calculus for compatible almost CR structures (also known as strictly pseudoconvex partially integrable almost CR structures) on contact manifolds, and as an application, express the first BGG invariant differential operator $D_0$ explicitly in some cases, i.e., for some tractor connections. An interesting outcome is the fact that the "modified" adjoint tractor connection $\tilde{\nabla}$ governing infinitesimal deformations of parabolic geometries generates what we call the CR Killing operator as its first BGG operator, and actually, it does not agree with the first BGG operator of the normal (unmodified) adjoint tractor connection. The relationship between the CR Killing operator and analysis of ACHE (asymptotically complex hyperbolic Einstein) metrics, or more specifically the CR obstruction tensor, is also discussed.

math.DG↗
arXiv2021

Canonical almost complex structures on ACH Einstein manifolds

On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler-Lagrange equation at Kähler-Einstein structures is given by the Dolbeault Laplacian acting on $(0,1)$-forms with values in the holomorphic tangent bundle. A deformation result of Einstein ACH metrics associated with critical almost complex structures for this variational problem is given. It is also shown that the asymptotic expansion of a critical almost complex structure is determined by the induced (possibly non-integrable) CR structure on the boundary at infinity up to a certain order.

math.DG↗
arXiv2019

Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an $L^2$ cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng-Yau metric, we establish the possibility of such a deformation when the dimension of the domain is larger than or equal to three.

math.DG↗
arXiv2018

A construction of Poincaré-Einstein metrics of cohomogeneity one on the ball

We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincaré-Einstein manifold of one lower dimension with an isolated conical singularity.

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↗
arXiv2015

GJMS operators, Q-curvature, and obstruction tensor of partially integrable CR manifolds

We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontrivial in dimension at least five. It is shown that its variation is given by the curvature-type quantity called the CR obstruction tensor, which is introduced in the author's previous work. Moreover, we consider the linearized CR obstruction operator. Based on a scattering-theoretic characterization, we discuss its relation to the CR deformation complex of integrable CR manifolds. The same characterization is also used to determine the Heisenberg principal symbol of the linearized CR obstruction operator.

math.DG↗
arXiv2015

Q-curvature of Weyl structures and Poincaré metrics

We study an asymptotic Dirichlet problem for Weyl structures on asymptotically hyperbolic manifolds. By the bulk-boundary correspondence, or more precisely by the Fefferman-Graham theorem on Poincaré metrics, this leads to a natural extension of the notion of Branson's $Q$-curvature to Weyl structures on even-dimensional conformal manifolds.

math.DG↗
arXiv2013

A GJMS construction for 2-tensors and the second variation of the total Q-curvature

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operators is used to describe the second variation of the total Q-curvature. An explicit formula for conformally Einstein manifolds is given in terms of the Lichnerowicz Laplacian of an Einstein representative metric.

math.DG↗
arXiv2011

Asymptotics of ACH-Einstein metrics

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equation if we allow logarithmic terms, and that a local CR-invariant tensor arises as the obstruction to the existence of a log-free solution. Some properties of this new CR invariant, the CR obstruction tensor, are discussed.

math.DG↗