A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry

Yıl 2024, Cilt: 17 Sayı: 2, 727 - 752, 27.10.2024

Bang-yen Chen , Majid Ali Choudhary , Nisar Mohammed , Mohd Danish Siddiqi

https://doi.org/10.36890/iejg.1526047

Öz

n Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is given in this review article, which also summarizes their historical evolution, core ideas, important findings, and applications. We investigate the complex interactions between curvature conditions and geometric inequalities as well as the several kinds of Ricci solitons, such as expanding, steady, and shrinking solitons. We also go over current developments, unresolved issues, and possible paths for further study in this fascinating area.

Anahtar Kelimeler

Ricci soliton, solitonic inequalities, $D$-homothetic deformation, statistical submersion, hyperbolic Ricci soliton, Hitchin-Thorpe inequality

Kaynakça

• [1] Azami, S., Fasihi-Ramandi, G.: Hyperbolic Ricci soliton on warped product manifolds. Filomat, 37(20), 6843–6853 (2023).
• [2] Bamler, R. H., Cifarelli, C., Conlon, R. J., Deruelle, A.: A new complete two-dimensional shrinking gradient Kähler Ricci soliton. Geometric and Functional Analysis, 1–16, (2024).
• [3] Basu, N., Bhattacharyya, A.: Conformal Ricci soliton in Kenmotsu manifold. Global Journal of Advanced Research on Classical and Modern Geometries, 4(1), 15–21 (2015).
• [4] Besse, A. L.: Einstein manifolds. Springer (2007).
• [5] Blaga, A. M.: Geometric solitons in a D-homothetically deformed Kenmotsu manifold. Filomat. 36(1), 175–186 (2022).
• [6] Blaga, A. M., Chen, B.-Y.: Gradient solitons on statistical manifolds. J. Geom. Phys. 164 , Paper No. 104195, 10 pp (2021).
• [7] Blaga, A. M., Crasmareanu, M.: Inequalities for gradient Einstein and Ricci solitons. Facta Universitatis, Series: Mathematics and Informatics, 351–356 (2020).
• [8] Blair, D. E.: Riemannian geometry of contact and symplectic manifolds. Progr. Math., vol. 203, Birkhäuser Boston, Inc., Boston, MA, (2010).
• [9] Borges, V., Tenenblat, K.: Ricci almost solitons on semi-Riemannian warped products. Mathematische Nachrichten 295(1), 22–43 (2022).
• [10] Brendle, S. Rotational symmetry of self-similar solutions to the Ricci flow. Inventiones Mathematicae, 194(3), 731–764 (2013).
• [11] Brendle, S.: Rotational symmetry of Ricci solitons in higher dimensions. Journal of Differential Geometry, 97(2), 191–214 (2014).
• [12] Bryant, R. L.: Ricci flow solitons in dimension three with SO(3)-symmetries. Department of Mathematics, Duke University (2005). https://services.math.duke.edu/∼bryant/3DRotSymRicciSolitons.pdf
• [13] Calin, C., Crasmareanu, M. η-Ricci solitons on Hopf hypersurfaces in complex space forms. Rev. Roumaine Math. Pures Appl, 57(1), 55–63 (2012).
• [14] Cao, H.-D.: On Harnack’s inequalities for the Kähler Ricci flow. Inventiones Mathematicae, 109(1), 247–263 (1992).
• [15] Cao, H.-D.: Existence of gradient Kähler Ricci solitons. In Elliptic and parabolic methods in geometry, pp. 1–16, AK Peters/CRC Press (1996).
• [16] Cao, H.-D.: Recent progress on Ricci solitons. ArXiv Preprint ArXiv:0908.2006 (2009).
• [17] Cao, H.-D., Cui, X.: Curvature estimates for four-dimensional gradient steady Ricci solitons. The Journal of Geometric Analysis, 30, 511–525 (2020).
• [18] Cao, H.-D., Liu, T., Xie, J.: Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio. Calculus of Variations and Partial Differential Equations, 62(2), Paper No. 48 (2023).
• [19] Cao, H.-D., Sesum, N.: A compactness result for Kähler Ricci solitons. Advances in Mathematics, 211(2), 794–818 (2007).
• [20] Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. Journal of Differential Geometry, 85(2), 175–186 (2010).
• [21] Carrillo, J., Ni, L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Communications in Analysis and Geometry, 17(4), 721–753 (2009).
• [22] Chan, P.-Y.: Curvature estimates for steady Ricci solitons. Transactions of the American Mathematical Society, 372(12), 8985–9008 (2019).
• [23] Cheeger, J., Ebin, D. G., Ebin, D. G.: Comparison theorems in Riemannian geometry. North-Holland publishing company Amsterdam (1975).
• [24] Chen, B.-Y.: On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms. Archiv der Mathematik, 74(2), 154–160 (2000).
• [25] Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatshefte für Mathematik 133(3), 177–195 (2001).
• [26] Chen, B.-Y.: Differential geometry of warped product manifolds and submanifolds. World Scientific, Hackensack, NJ (2017).
• [27] Chen, B.-Y., Deshmukh, S.: Geometry of compact shrinking Ricci solitons. Balkan J. Geom. Appl, 19(1), 13-21 (2014).
• [28] Chen, B.-Y., Siddiqi, M. D., Siddiqui, A. N.: On Ricci-Bourguignon solitons for statistical submersions. Bull. Korean Math. Soc. (In press) (2024).
• [29] Cho, J. T., Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form. Tohoku Mathematical Journal, 61(2), 205–212 (2009).
• [30] Choudhary, A. M., Siddiqi, M. D., Bahadır, O., Uddin, S.: Hypersurfaces of metallic Riemannian manifolds as k-almost Newton-Ricci solitons. Filomat, 37(7), 2187–2197 (2023).
• [31] Choudhary, M. A., Blaga, A. M.: Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms. Journal of Geometry, 112(2), 26 (2021).
• [32] Choudhary, M. A., Das, L. S., Siddiqi, M. D., Bahadır, O.: Generalized Wintgen Inequalities for (ϵ)-Para Sasakian Manifold. In Geometry of Submanifolds and Applications (pp. 133–145). Springer (2024).
• [33] Choudhary, M. A., Khan, M. N. I., Siddiqi, M. D.: Some basic inequalities on para Sasakian manifold. Symmetry, 14(12), 2585 (2022).
• [34] Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow (Vol. 77). American Mathematical Society, Science Press (2023).
• [35] Deruelle, A.: Asymptotic estimates and compactness of expanding gradient Ricci solitons. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 17(2), 485–530 (2017).
• [36] Deshmukh, S., Alsodais, H., Bin Turki, N.: Some results on Ricci almost solitons. Symmetry, 13(3), 430 (2021).
• [37] Donaldson, S. K., Kronheimer, P. B.: The geometry of four-manifolds. Oxford university press (1997).
• [38] Einstein, A.: Die feldgleichungen der gravitation. Sitzungsberichte Der Königlich Preußischen Akademie Der Wissenschaften, 844–847 (1915).
• [39] Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler Ricci solitons. Journal of Differential Geometry, 65(2), 169–209 (2003).
• [40] Fermi, E., Pasta, P., Ulam, S., Tsingou, M.: Studies of the nonlinear problems [Techreport]. Los Alamos National Lab., Los Alamos, NM, USA (1955).
• [41] Fischer, A. E.: An introduction to conformal Ricci flow. Classical and Quantum Gravity, 21(3), S171-S218 (2004).
• [42] Ganguly, D., Bhattacharyya, A.: A study on conformal Ricci solitons in the framework of (LCS)n-manifolds. Ganita, 70(2), 201–216 (2020).
• [43] Ganguly, D., Dey, S., Ali, A., Bhattacharyya, A.: Conformal Ricci soliton and Quasi-Yamabe soliton on generalized Sasakian space form. Journal of Geometry and Physics, 169, Paper No. 104339 (2021).
• [44] Gardner, C. S., Greene, J. M., Kruskal, M. D., Miura, R. M.: Method for solving the Korteweg-deVries equation. Physical Review Letters, 19(19), 1095 (1967).
• [45] Hamilton, R. S.: Three-manifolds with positive Ricci curvature. Journal of Differential Geometry, 17(2), 255–306 (1982).
• [46] Hamilton, R. S.: Four-manifolds with positive curvature operator. Journal of Differential Geometry, 24(2), 153–179 (1986).
• [47] Hamilton, R. S.: The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math., 71, 237–261 (1988).
• [48] Hamilton, R.: The formations of singularities in the Ricci Flow. Surveys in Differential Geometry, 2(1), 7–136 (1993).
• [49] Hawking, S. W., Ellis, G. F.: The large scale structure of space-time. Cambridge University Press (2023).
• [50] Hitchin, N.: Compact four-dimensional Einstein manifolds. Journal of Differential Geometry, 9(3), 435–441 (1974).
• [51] Hretcanu, C.-E., Crasmareanu, M.: Metallic structures on Riemannian manifolds. Revista de la Unió Matemática Argentina, 54(2), 15–27 (2013).
• [52] Jost, J.: Riemannian geometry and geometric analysis. 6th edition, Springer (2011).
• [53] Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics. In Recent topics in differential and analytic geometry (pp. 327–337). Elsevier (1990).
• [54] Kröncke, K.: Stability and instability of Ricci solitons. Calculus of Variations and Partial Differential Equations, 53, 265–287 (2015).
• [55] Li, P., Yau, S. T.: On the parabolic kernel of the Schrödinger operator. Acta Mathematica, 156, 153–201 (1986).
• [56] Li, Y., Srivastava, S. K., Mofarreh, F., Kumar, A., Ali, A.: Ricci soliton of CR-warped product manifolds and their classifications, Symmetry, 15, Paper No. 976, 14 pp (2023).
• [57] Li, Y. L., Ganguly, D., Dey, S., Bhattacharyya, A.: Conformal η-Ricci solitons within the framework of indefinite Kenmotsu manifolds. AIMS Math, 7(4), 5[7]8–5430 (2022).
• [58] Miura, C., Ablowitz, M., Kaup, D., Newell, A., Segur, H.: Addendum: A classical perturbation theory [J. Math. Phys. 18, 110 (1977)]. Stud. Appl. Math, 53(249), 249 (1974).
• [59] Morgan, J. W., Tian, G.: Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, vol. 3. American Mathematical Society (2007).
• [60] Munteanu, O., Sung, C.-J. A., Wang, J.: Poisson equation on complete manifolds. Advances in Mathematics, 348, 81–145 (2019).
• [61] Munteanu, O., Wang, J.: Geometry of shrinking Ricci solitons. Compositio Mathematica, 151(12), 2273–2300 (2015).
• [62] O’Neill, B.: The fundamental equations of a submersion. Michigan Mathematical Journal, 13(4), 459–469 (1966).
• [63] O’Neill, B.: Semi-Riemannian geometry with applications to relativity.Academic Press, New York, NY (1983).
• [64] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. ArXiv Preprint Math/0211159 (2002).
• [65] Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, ArXiv Preprint Math.DG/0307245, 16 (2003).
• [66] Perelman, G.: Ricci flow with surgery on three-manifolds. ArXiv Preprint Math/0303109 (2003).
• [67] Poincaré, H.: Sur la dynamique de l’électron. Comptes rendus de l’Académie des Sciences, 140, 1504–1508 (1905).
• [68] Rimoldi, M., Veronelli, G.: Extremals of log Sobolev inequality on non-compact manifolds and Ricci soliton structures. Calculus of Variations and Partial Differential Equations, 58(2), Paper No. 66, 26pp (2019).
• [69] Schoen, R. M., Yau, S. T.: Lectures on differential geometry. International Press, Cambridge, MA (1994).
• [70] Seshadri, H.: Weyl curvature and the Euler characteristic in dimension four. Differential Geometry and Its Applications, 24(2), 172–177 (2006).
• [71] Siddiqi, M. D.: Conformal η-Ricci solitons in δ-Lorentzian Trans Sasakian manifolds. International Journal of Maps in Mathematics, 1(1), 15–34 (2018).
• [72] Streets, J., Tian, G.: Symplectic curvature flow. Journal für Die Reine Und Angewandte Mathematik, 696, 143–185 (2014).
• [73] Tanno, S.: Variational problems on contact Riemannian manifolds. Transactions of the American Mathematical Society, 314(1), 349–379 (1989).
• [74] Thorpe, J. A.: Some remarks on the Gauss-Bonnet integral. Journal of Mathematics and Mechanics, 18(8), 779–786 (1969).
• [75] Wadati, M.: The modified Korteweg-de Vries equation. Journal of the Physical Society of Japan, 34(5), 1289–1296 (1973).
• [76] Wadati, M.: Introduction to solitons. Pramana, 57, 841–847 (2001).
• [77] Wadati, M., Toda, M.: The exact N-soliton solution of the Korteweg-de Vries equation. Journal of the Physical Society of Japan, 32(5), 1[7]3–1411 (1972).
• [78] Wang, X.-J., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Advances in Mathematics, 188(1), 87–103 (2004).
• [79] Yano, K., Kon, M.: Structures on manifolds. World scientific, Singapore (1985).
• [80] Zabusky, N. J., Kruskal, M. D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Physical Review Letters, 15(6), 2[7] (1965).
• [81] Zakharov, V. E., Faddeev, L. D.: Korteweg–de Vries equation: A completely integrable Hamiltonian system. Funktsional’nyi Analiz i Ego Prilozheniya, 5(4), 18–27 (1971).
• [82] Zakharov, V., Shabat, A.: Interaction between solutions in a stable medium. Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki, 64(5), 1627–1639 (1973).
• [83] Zhang, Q. S.: Extremal of Log Sobolev inequality and W entropy on noncompact manifolds. Journal of Functional Analysis, 263(7), 2051–2101 (2012).
• [84] Zhang, X.: Compactness theorems for gradient Ricci solitons. Journal of Geometry and Physics, 56(12), 2481–2499 (2006).
• [85] Zhang, Z.-H.: On the completeness of gradient Ricci solitons. Proceedings of the American Mathematical Society, 137(8), 2755–2759 (2009).