Function Theory on Manifolds Which Possess a Pole

On the Bergman metric of pseudoconvex domains in a complex projective space

“…We mention that there are a lot of works concerning the completeness of β M (cf. [4], [5], [6], [12], [13], [14], [15], [17], [20]). On the other hand, Diederich and Ohsawa [10] proved that the Bergman distance for a bounded C 2 pseudoconvex domain in C n has a lower bound of a constant multiple log | log δ Ω |.…”

“…By the Hessian comparison theorem (cf. [12]), there is a positive constant C 1 independent of y such that ∂∂φ y (x) ≥ −C 1 ds 2 F S . From now on we assume r 0 = 1 for the sake of simplicity.…”

Interpolation in non-positively curved Kähler manifolds

“…We will state the comparison theorem for the Hessian (see [4] Theorem A page 19) in a form suitable for our purpose. The proof of it relies on a precise form of the formula for second order variation of arc length, and on a relation between real and complex Hessian on Kähler manifolds (see [11] page 102).…”

“…In the next section, we recall some comparison theorem in Riemannian geometry from [4] and [10]. In section 2, we prove that under some assumption relating the complex structure and the metric structure uniformly on Λ, the uniform separation is necessary.…”

On the Liouville Theorem for Harmonic Maps

“…The following Bochner type formula is also well known arbitrarily to vector fields in a neighborhood of q. DXY denotes the covariant derivative of the Riemannian connection of N. It is easy to check D2<j> is a tensor and ö2«j>(/" fj) = <Pij. We also have A"<¿> = 2, <í»", Let y be a strictly convex function defined in a neighborhood of a point q -u(p) E N, p E M. Suppose \(q) is the lower bound of the eigenvalues of the Hessian tensor (<f>,y) at q, then we have (2) A«(*°«)t= 2 <t>u"iaUja>H^)e(u).…”