×

The isoperimetric problem in the 2-dimensional Finsler space forms with \(k=0\). II. (English) Zbl 1476.53049

Authors’ abstract: This paper is a continuation of the second author’s previous work [Int. J. Math. 30, No. 1, Article ID 1950005, 18 p. (2019; Zbl 1408.53032)]. We investigate the isoperimetric problem in the 2-dimensional Finsler space form \((F_B, B^2(1))\) with \(k=0\) by using the Holmes-Thompson area and prove that the circle centered at the origin achieves the local maximum area of the isoperimetric problem.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds

Citations:

Zbl 1408.53032
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bolza, Oskar, Lectures on the calculus of variations, ix+271 pp. (1961), 2nd ed. Chelsea Publishing Co., New York · Zbl 1422.49001
[2] Cheng, Xinyue; Shen, Zhongmin, A class of Finsler metrics with isotropic \(S\)-curvature, Israel J. Math., 169, 317-340 (2009) · Zbl 1165.53016 · doi:10.1007/s11856-009-0013-1
[3] Hestenes, M. R., A sufficiency proof for isoperimetric problems in the calculus of variations, Bull. Amer. Math. Soc., 44, 10, 662-667 (1938) · Zbl 0019.35301 · doi:10.1090/S0002-9904-1938-06838-3
[4] Li, Benling, On the classification of projectively flat Finsler metrics with constant flag curvature, Adv. Math., 257, 266-284 (2014) · Zbl 1294.53069 · doi:10.1016/j.aim.2014.02.022
[5] Mo, Xiaohuan; Zhu, Hongmei, On a class of projectively flat Finsler metrics of negative constant flag curvature, Internat. J. Math., 23, 8, 1250084, 14 pp. (2012) · Zbl 1257.58009 · doi:10.1142/S0129167X1250084X
[6] Osserman, Robert, The isoperimetric inequality, Bull. Amer. Math. Soc., 84, 6, 1182-1238 (1978) · Zbl 0411.52006 · doi:10.1090/S0002-9904-1978-14553-4
[7] \'{A}lvarez Paiva, J. C.; Thompson, A. C., Volumes on normed and Finsler spaces. A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ. 50, 1-48 (2004), Cambridge Univ. Press, Cambridge · Zbl 1288.30051 · doi:10.4171/prims/123
[8] Zhou, Linfeng, Projective spherically symmetric Finsler metrics with constant flag curvature in \(R^n\), Geom. Dedicata, 158, 353-364 (2012) · Zbl 1243.53044 · doi:10.1007/s10711-011-9639-3
[9] Zhou, Linfeng, The isoperimetric problem in the 2-dimensional Finsler space forms with \(k=0\), Internat. J. Math., 30, 1, 1950005, 18 pp. (2019) · Zbl 1408.53032 · doi:10.1142/S0129167X19500058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.