Inspired by the halfspace theorem for minimal surfaces in $\mathbb {R}^3$ of Hoffman–Meeks, the halfspace theorem of Rodriguez–Rosenberg, and the classical cone theorem of Omori in $\mathbb {R}^n$, we derive new non-existence results for proper harmonic maps into perturbed cones in $\mathbb {R}^n$, horospheres in $\mathbb {H}^n$, culminating in a generalization of Omori’s theorem in arbitrary Riemannian manifolds. The technical tool proved here extends the foliated Sampson’s maximum principle, initially developed in the first author’s Ph.D. thesis, to a non-compact setting.

We give some effectivity results in birational geometry. We provide an upper bound on the rational constant in Rationality Theorem in terms of certain intersection numbers, under an additional condition on the variety that it admits a divisorial contraction. One consequence is an explicit bound on the number of certain extremal rays. Our main result tries to construct from a given set of ample divisors Hj on X with their intersection numbers bi, a certain set of ample divisors Lj on X' or X+ where X' or X+ arises from a contraction or a flip, such that the corresponding intersection numbers of Lj are uniformly bounded in terms of bi and the index of X. This gives a bound on the projective degree of a minimal model in special case.