• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.3
• /
• pp.457-467
• /
• 2011

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.4
• /
• pp.369-378
• /
• 2021

In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L²-norm of the traceless second fundamental form. If M is weakly stable, then λ₁(M) is bounded above by n² + O(n^2+s) for arbitrary s > 0.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.319-326
• /
• 2003

In this paper, we obtain the optimal condition of the curvature bounds guaranteeing that Euclidean cones over Aleksandrov spaces of curvature bounded above preserve the curvature bounds, by considering the Euclidean cone CS$_{r}$ $^{n}$ over n-dimensional sphere S$_{r}$ $^{n}$ of radius r. More precisely, we show that for r<1, the Euclidean cone CS$_{r}$ $^{n}$ of S$_{r}$ $^{n}$ is a CBB(0) space, but not a CBA($textsc{k}$)-space for any real $textsc{k}$$\in$R.