Mitsuhiro Itoh

ALMOST KÄHLER 4-MANIFOLDS, L2-SCALAR CURVATURE FUNCTIONAL AND SEIBERG–WITTEN EQUATIONS

We show in this paper by applying the Seiberg–Witten theory developed by Taubes and LeBrun that a compact almost Kahler–Einstein 4-manifold of negative scalar curvature s is Kahler–Einstein if and only if the L2-norm satisfies ∫Ms2dv=32π2(2χ+3τ)(M). The Einstein condition can be weakened by the topological condition (2χ+3τ)(M)>0.

Contact metric 5-manifolds, CR twistor spaces and integrability

The CR twistor space is defined over a contact metric 5-manifold M. Like the 4-dim twistor theory, the integrability of the almost CR twistor structure is discussed in terms of the Weyl conformal curvature and also the scalar curvature of M.

The modified Yamabe problem and geometry of modified scalar curvatures

Study of modified scalar curvatures and the modified Yamabe problem is given. An existence theorem of type of Trudinger and Aubin is obtained. Interpreting Isotropic curvature as a certain modified scalar curvature we can extend the four-dimensional sphere theorem of [23]. Another modified scalar curvature is used in solving open problems concerning the conjecture of self-dual Einstein 4-manifold and the Goldberg conjecture, both being under negative Ricci condition.

Generalized magnetic monopoles over contact manifolds

A generalization of magnetic monopoles is given over an odd dimensional contact manifold and we discuss whether the Yang–Mills–Higgs functional attains at generalized monopoles the absolute minimal value, the topological invariant.

Geometry of Fisher Information Metric and the Barycenter Map

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold \(X\). We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold \(X\) and its ideal boundary \(\partial X\)—for a given homeomorphism \(\Phi\) of \(\partial X\) find an isometry of \(X\) whose \(\partial X\)-extension coincides with \(\Phi\)—is investigated in terms of the barycenter map.

Information Geometry of Barycenter Map

Using barycenter of the Busemann function we define a map, called the barycenter map from a space \(\mathcal{P}^{+}\) of probability measures on the ideal boundary ∂ X to an Hadamard manifold X. We show that the space \(\mathcal{P}^{+}\) carries a fibre space structure over X from a viewpoint of information geometry. Following the idea of [7, 9] and [8] we present moreover a theorem which states that under certain hypotheses of information geometry a homeomorphism Φ of ∂ X induces, via the push-forward for probability measures, an isometry of X whose ∂ X-extension coincides with Φ.

Information geometry of Busemann-barycenter for probability measures

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidites des espaces localement symetriques de coubure strictement negative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostows rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

Fisher information geometry of the barycenter map

We report Fisher information geometry of the barycenter map associated with Busemann function Bθ of an Hadamard manifold X and present its application to Riemannian geometry of X from viewpoint of Fisher information geometry. This report is an improvement of [I-Sat’13] together with a fine investigation of the barycenter map.

The Serre duality theorem for a non-compact weighted CR manifold

It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented.

Poincaré Bundle and Chern Classes

This chapter discusses the positivity of the first Chern class ; it is the first Chern class of the holomorphic tangent bundle , and the bundle can be regarded as an index bundle arising from the Dolbeault operator coupling to all anti-self-dual connections on P . The chapter presents a scenario where ( X , g ) is a compact Kahler surface and P an SU (2) bundle over X of index k = c 2 ( P × p C 2 ). It also focuses on the Hodge structure and the determinant bundle.