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Yoshihiko Mitsumatsu

Publications and source records attributed to Yoshihiko Mitsumatsu.

15 recordsLinked to original sources
arXiv2026

Real analytic lift of foliations of Thurston and Tsuboi

Thurston constructed codimension one foliations on $S^3$ thereby proved that the homomorphism $gv: π_3(B\overlineΓ^\infty_1)\rightarrow \mathbb{R}$ induced by the Godbillon-Vey invariant is surjective. By another real analytic construction, he proved that the homomorphism $gv: H_3(B\overlineΓ^ω_1)\rightarrow \mathbb{R}$ is also surjective where $B\overlineΓ^ω_1$ is a $K(π,1)$ space by Haefliger. Tsuboi proved that the former surjection splits so that $π_3(B\overlineΓ^\infty_1)= \mathbb{R}\oplus \mathrm{Ker}\,gv$. He further showed that the subgroup of $H_3(B\overlineΓ^\infty_1;\mathbb{Z})$ generated by all the Thurston's constructions coincides with his direct summand $\mathbb{R}$. In this paper, we prove that Thurston's second surjection splits and also that the subgroup of $H_3(B\overlineΓ^ω_1;\mathbb{Z})$ generated by all the Thurston's cycles is equal to our direct summand $\mathbb{R}$ which is a lift of Tsuboi's one. To show this, we modify the arguments of Thurston and Tsuboi by replacing Reeb components with a real analytic construction. We prove certain {\it uniqueness} of them by showing acyclicity of the affine group in the Haefliger group $π_1(B\overlineΓ^ω_1)$. We also prove the existence of a new kind of characteristic class of foliations in $H^4(B\overlineΓ^ω_1;\mathbb{Z})$.

math.GT↗
arXiv2026

Lefschetz fibrations on the Milnor fibers of cusp and simple elliptic singularities

We show that the total space of the Milnor fibration associated with any cusp or simple elliptic singularity in complex three variables admits an $S^1$-parametric genus-one Lefschetz fibration structure over the $2$-disk. As a consequence, we demonstrate that the Lawson type foliations on $S^5$ associated with such singularities can be regarded as the pullback of the Reeb foliation on $S^3$. This enables us to provide an alternative proof of a previous result by the third author, which states that every Lawson type foliation admits a leafwise symplectic structure. Also we see that a pair of such Milnor fibers can be glued together along boundary into a closed oriented 4-manifold exactly when the pair corresponds to one of the ten extended strange duality pairs among the cusp singularities. This gluing is compatible with the Lefschetz fibrations and the resultant 4-manifold is diffeomrphic to a K3 surface.

math.GT↗
arXiv2025

On the existence of critical compatible metrics on contact $3$-manifolds

We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact $3$-manifolds. More precisely, we show that a contact $3$-manifold $(M,α)$ admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is $C^\infty$-conjugate to an algebraic Anosov flow modeled on $\widetilde{SL}(2, \mathbb R)$. In particular, this yields a complete topological classification of compact $3$-manifolds that admit critical compatible metrics. As a corollary we prove that no contact structure on $\mathbb{T}^3$ admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is tight.

math.DG↗
arXiv2023

Remarks on flat $S^1$-bundles, $C^\infty$ vs $C^ω$

We describe low dimensional homology groups of $\mathrm{Diff}^δ_+S^1$ in terms of Haefliger's classifying space $B\overlineΓ_1$ by applying a theorem of Thurston. Then we consider the question whether some power of the rational Euler class vanishes for real analytic flat $S^1$-bundles. We show that if it occurs, then the homology group of $\mathrm{Diff}_+^{ω,δ} S^1$ should contain two kinds of many torsion classes which vanish in $\mathrm{Diff}^δ_+S^1$. This is an informal note on our discussions about the above question.

math.GT↗
arXiv2022

Modification of Convex Ends to Cylindrical and Symplectic Foliations

We show that the natural symplectic structure on the Milnor fiber of an isolated singularity in complex three variables whose link fibers over the circle can be modified into one which is cylindrical at the end. As a consequence we see that the foliation of codimension one on S^5 which is adapted to the Milnor open book of S^5 associated with such a singularity admits a leafwise symplectic structure. The modification also enables us to construct certain closed symplectic 4-manifolds.

math.SG↗
arXiv2018

Derivatives of flat functions

We remark that there is no smooth function $f(x)$ on $[0, 1]$ which is flat at $0$ such that the derivative $f^{(n)}$ of any order $n\geq 0$ is positive on $(0,1]$. Moreover, the number of zeros of the $n$-th derivative $f^{(n)}$ grows to the infinity and the zeros accumulate to $0$ when $n \to \infty$.

math.GM↗
arXiv2018

Geometry and dynamics of Engel structures

The aim of this paper is to extend basic understanding of Engel structures through developing geometric constructions which are canonical to a certain degree and the dynamics of Cauchy characteristics in the transverse spaces which may exhibit elliptic, parabolic, or hyperbolic natures in typical cases.

math.DG↗
arXiv2016

Reeb components of leafwise complex foliations and their symmetries I

We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.

math.GT↗
arXiv2016

Reeb components of leafwise complex foliations and their symmetries III

The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components obtained by the Hopf construction, we have an almost complete description of the groups of leafwise holomorphic smooth automorphisms.

math.GT↗
arXiv2016

Reeb components with complex leaves and their symmetries I : The automorphism groups and Schröder's equation on the half line

We review the standard Hopf construction of Reeb components with leafwise complex structure and determine the group of leafwise holomorphic smooth automorphisms for tame Reeb components in the case of complex leaf dimension one. For this, we solve the Schröder type functional equation on the half line for expanding diffeomorphism. As a result, we see that the automorphism group of one with trivial linear holonomy on the boundary contains an infinite dimensional vector space, while in the case of non-trivial linear holonomy the group is of finite dimensional.

math.GT↗