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I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded one-sided tubular neighborhood of a geodesic in the hyperbolic plane (see https://mathoverflow.net/a/149884/1345), seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface (or rather family of surfaces), which are isometrically embedded one-sided tubular neighborhoods of a geodesic in the hyperbolic plane (see https://mathoverflow.net/a/149884/1345), seems to do the trick since they contain arbitrarily large disks in the hyperbolic plane by varying the parameter so that the radius of the tubular neighborhood grows. See Dini's Surface at the Geometry Center.

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded one-sided tubular neighborhood of a geodesic in the hyperbolic plane (see https://mathoverflow.net/a/149884/1345), seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

alt text http://paulbourke.net/geometry/dini/dini3.gif

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.