Generalized Maximal Diameter Theorems

Year 2024, , 199 - 206, 23.04.2024

James Hebda Yutaka Ikeda

https://doi.org/10.36890/iejg.1384669

Abstract

We prove Maximal Diameter Theorems for pointed Riemannian manifolds which are compared to surfaces of revolution with weaker radial attraction.

Keywords

Maximal diameter theorem, maximal perimeter theorem, weaker radial attraction

References

• [1] Boonnam, N.: A generalized maximal diameter sphere theorem. Tohoku Math. J. 71 145-155 (2019).
• [2] Gluck H., Singer, D.: Scattering of geodesic fields II. Annals of Math. 110 205-225 (1979).
• [3] Hebda, J., Ikeda, Y.: Replacing the Lower Curvature Bound in Toponogov’s Comparison Theorem by a Weaker Hypothesis. Tohoku Math. J. 69 305-320 (2017).
• [4] Hebda, J., Ikeda, Y.: Necessary and Sufficient Conditions for a Triangle Comparison Theorem. Tohoku Math. J. 74 329-364 (2022).
• [5] Innami, N., K. Shiohama, K., Uneme, Y.: The Alexandrov–Toponogov Comparison Theorem for Radial Curvature. Nihonkai Math. J. 24 57-91 (2013).
• [6] Itokawa, Y., Machigashira, Y., Shiohama, K.:Generalized Toponogov’s Theorem for manifolds with radial curvature bounded below. Contemporary Mathematics. 332 121-130 (2003).
• [7] Shiohama, K., and Tanaka, M.: Compactification and maximal diameter theorem for noncompact manifolds with curvature bounded below. Mathematische Zeitschrift. 241 341-351 (2002).
• [8] Sinclair, R., Tanaka, M.: The cut locus of a sphere of revolution and Toponogov’s comparison theorem. Tohoku Math. J. 59 379-399 (2007).
• [9] Soga, T.: Remarks on the set of poles on a pointed complete surface. Nihonkai Math. J. 22 23-37 (2011).
• [10] Tanaka, M.: On a characterization of a surface of revolution with many poles. Mem. Fac. Sci. Kyushu Univ. Ser. A 46 251-268(1992).