2019
Том 71
№ 5

# Nguyen Doan Tuan

Articles: 3
Brief Communications (English)

### On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1576–1583

In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation Kk > 0, then $d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}$ for any point mM. In the case where dim M = 2, the Gaussian curvature K satisfies the relation Kk ≥ 0, and γ is of length l, we get Vol (M, g) ≤ $\frac{{2l}}{{\sqrt k }}$ if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.

Article (English)

### A Higher-Dimensional Version of the Brody Reparametrization Lemma

Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1369-1377

We prove a generalization of the Brody reparametrization lemma.

Brief Communications (English)

### On the degree of holomorphic mappings from an annulus into an annulus

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1135–1138

We establish an estimate for the degree of a holomorphic mapping f : K 1 → K 2 (here, K 1 and K 2 are doubly-connected domains) in terms of the modulus of a family of curves in K 2. This estimate generalizes a result obtained by Kobayashi.