Minimal networks on balls and spheres for almost standard metrics. (English) Zbl 07919127
Summary: We study the existence of minimal networks in the unit sphere \(\mathbf{S}^d\) and the unit ball \(\mathbf{B}^d\) of \(\mathbf{R}^d\) endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of \(\theta\)-networks in \(\mathbf{S}^d\) and triods in \(\mathbf{B}^d\), jointly with the Lusternik-Schnirelmann category.
MSC:
| 53C22 | Geodesics in global differential geometry |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
Keywords:
minimal networks; geodesic nets; finite-dimensional reduction; Lusternik-Schnirelmann categoryReferences:
| [1] | Ambrosetti, A.; Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems, 2007, Cambridge: Cambridge University Press, Cambridge · Zbl 1125.47052 · doi:10.1017/CBO9780511618260 |
| [2] | Berger, MS; Bombieri, E., Poincaré’s: on isoperimetric problem for simple closed geodesics, J. Funct. Anal., 42, 3, 274-295, 1981 · Zbl 0469.58007 · doi:10.1016/0022-1236(81)90091-4 |
| [3] | Birkhoff, GD, Dynamical systems with two degrees of freedom, Trans. Am. Math. Soc., 18, 2, 199-300, 1917 · JFM 46.1174.01 · doi:10.2307/1988861 |
| [4] | Bishop, RL, There is more than one way to frame a curve, Am. Math. Month., 82, 246-251, 1975 · Zbl 0298.53001 · doi:10.2307/2319846 |
| [5] | Cheng, HY, Nets, stable geodesic, in convex hypersurfaces, J. Geom. Anal., 34, 2, 56, 2024 · Zbl 1534.53047 · doi:10.1007/s12220-023-01489-2 |
| [6] | Chambers, GR; Liokumovich, Y.; Nabutovsky, A.; Rotman, R., Geodesic nets on non-compact Riemannian manifolds, J. Reine Angew. Math., 799, 287-303, 2023 · Zbl 07691725 · doi:10.1515/crelle-2023-0028 |
| [7] | Cornea, O.; Lupton, G.; Oprea, J.; Tanré, D., Lusternik-Schnirelmann Category, 2003, Providence: Mathematical Surveys and Monographs, American Mathematical Society, Providence · Zbl 1032.55001 · doi:10.1090/surv/103 |
| [8] | Chodosh, O.; Mantoulidis, C., The p-widths of a surface, Publ. Math. Inst. Hautes Études Sci., 137, 245-342, 2023 · Zbl 1533.58015 · doi:10.1007/s10240-023-00141-7 |
| [9] | Croke, CB, Poincaré’s problem and the length of the shortest closed geodesic on a convex hypersurface, J. Differ. Geom., 17, 4, 595-634, 1983 · Zbl 0501.53031 |
| [10] | Etayo, F., Rotation minimizing vector fields and frames in Riemannian manifolds, Geom. Algebra Appl., 8, 91-100, 2016 · Zbl 1372.53036 |
| [11] | Freire, A., The existence problem for Steiner networks in strictly convex domains, Arch. Ration. Mech. Anal., 200, 2, 361-404, 2011 · Zbl 1271.90095 · doi:10.1007/s00205-011-0414-2 |
| [12] | Grayson, MA, Shortening embedded curves, Ann. Math., 129, 2, 71-111, 1989 · Zbl 0686.53036 · doi:10.2307/1971486 |
| [13] | Hatcher, A.: Vector bundles and k-theory, (2017), https://pi.math.cornell.edu/ hatcher/VBKT/VB.pdf |
| [14] | Heppes, A., Isogonale sphärische Netze, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 7, 41-48, 1964 · Zbl 0127.37601 |
| [15] | Hass, J.; Morgan, F., Geodesic nets on the \(2\)-sphere, Proc. Am. Math. Soc., 124, 12, 3843-3850, 1996 · Zbl 0871.53038 · doi:10.1090/S0002-9939-96-03492-2 |
| [16] | Hass, J.; Morgan, F., Geodesics and soap bubbles in surfaces, Math. Z., 223, 185-196, 1996 · Zbl 0865.53009 · doi:10.1007/PL00004560 |
| [17] | Ivanov, AO; Ptitsyna, IV; Tuzhilin, AA, Classification of closed minimal nets on two-dimensional flat tori, Rus. Math. Surv., 183, 12, 3-44, 1992 · Zbl 0818.90125 · doi:10.1070/SM1994v077n02ABEH003448 |
| [18] | Ivanov, AO; Tuzhilin, AA, Minimal networks: a review, Stud. Syst. Decis. Control, 69, 43-80, 2016 · Zbl 1359.05117 · doi:10.1007/978-3-319-40673-2_4 |
| [19] | Ivanov, AO; Tuzhilin, AA, Minimal Networks, 1994, Boca Raton: CRC Press, Boca Raton · Zbl 0842.90116 |
| [20] | Iwase, N., Lusternik-Schnirelmann category of a sphere-bundle over a sphere, Topology, 42, 3, 701-713, 2003 · Zbl 1035.55004 · doi:10.1016/S0040-9383(02)00026-5 |
| [21] | Klingenberg, WPA, Poincaré’s closed geodesic on a convex surface, Trans. Am. Math. Soc., 356, 2545-2556, 2004 · Zbl 1045.53005 · doi:10.1090/S0002-9947-04-03444-0 |
| [22] | Liokumovich, Y.; Staffa, B., Generic density of geodesic nets, Selecta Math. (N.S.),, 14, 30, 2024 · Zbl 1541.53054 · doi:10.1007/s00029-023-00901-7 |
| [23] | Lusternik, L.; Schnirelmann, L., Existence de trois géodésiques fermées sur toute surfaces de genre 0, C. R. Acad. Sci. Paris, 188, 534-536, 1929 · JFM 55.0316.01 |
| [24] | Martelli, B.; Novaga, M.; Pluda, A.; Riolo, S., Spines of minimal length, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17, 3, 1067-1090, 2017 · Zbl 1396.57031 |
| [25] | Morgan, F., Size-minimizing rectifiable currents, Invent. Math., 96, 2, 333-348, 1989 · Zbl 0645.49024 · doi:10.1007/BF01393966 |
| [26] | Morgan, F., Soap bubbles in \({\bf R}^2\) and in surfaces, Pacific J. Math., 162, 2, 347-361, 1994 · Zbl 0820.53002 · doi:10.2140/pjm.1994.165.347 |
| [27] | Nishimoto, T., On the Lusternik-Schnirelmann category of Stiefel manifolds, Topol. Appl., 154, 9, 1956-1960, 2007 · Zbl 1124.55003 · doi:10.1016/j.topol.2007.02.002 |
| [28] | Nabutovsky, A.; Parsch, F., Geodesic nets: some examples and open problems, Exp. Math., 32, 1, 1-25, 2023 · Zbl 1519.53031 · doi:10.1080/10586458.2020.1743216 |
| [29] | Nabutovsky, A.; Rotman, R., Shapes of geodesic nets, Geom. Topol., 11, 1225-1254, 2007 · Zbl 1134.53018 · doi:10.2140/gt.2007.11.1225 |
| [30] | Poincaré, H., Sur les lignes géodésiques des surfaces convexes, Trans. Am. Math. Soc., 6, 3, 237-274, 1905 · JFM 36.0669.01 · doi:10.2307/1986219 |
| [31] | Rotman, R., The length of a shortest geodesic net on a closed Riemannian manifold, Topology, 46, 4, 343-356, 2007 · Zbl 1125.53034 · doi:10.1016/j.top.2006.10.003 |
| [32] | Staffa, B.: Bumpy metrics theorem for geodesic nets, (2023), Available at arXiv:2107.12446 |
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.
