Abstract
A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly −1. Finally, we present some preliminary results in higher dimensions and six potential directions for future work.
Similar content being viewed by others
References
Aiex, N.S.: Index estimate of self-shrinkers in \(\mathbb{R}^3\) with asymptotically conical ends. Proc. Am. Math. Soc. 147(2), 799–809 (2019). https://doi.org/10.1090/proc/14306
Angenent, S.B.: Shrinking doughnuts. In: Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), Progress in Nonlinear Differential Equations and Their Applications, vol. 7, pp. 21–38. Birkhäuser Boston, Boston (1992)
Barrett, J.W., Deckelnick, K., Nüunberg, R.: A finite element error analysis for axisymmetric mean curvature flow (2019). arXiv:1911.05398
Berchenko-Kogan, Y.: The entropy of the angenent torus is approximately 1.85122. Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1583616
Berchenko-Kogan, Y.: Numerically computing the index of mean curvature flow self-shrinkers (2020). arXiv:2007.06094
Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. Math. (2) 175(2), 755–833 (2012). https://doi.org/10.4007/annals.2012.175.2.7
Colding, T.H., Minicozzi II, W.P., Pedersen, E.K.: Mean curvature flow. Bull. Am. Math. Soc. (N. S.) 52(2), 297–333 (2015). https://doi.org/10.1090/S0273-0979-2015-01468-0
Drugan, G.: An immersed \(S^2\) self-shrinker. Trans. Am. Math. Soc. 367(5), 3139–3159 (2015). https://doi.org/10.1090/S0002-9947-2014-06051-0
Drugan, G., Kleene, S.J.: Immersed self-shrinkers. Trans. Am. Math. Soc. 369(10), 7213–7250 (2017). https://doi.org/10.1090/tran/6907
Drugan, G., Lee, H., Nguyen, X.H.: A survey of closed self-shrinkers with symmetry. Results Math. 73(1), Art. 32, 32 (2018). https://doi.org/10.1007/s00025-018-0763-3
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285–299 (1990)
Impera, D., Rimoldi, M., Savo, A.: Index and first Betti number of \(f\)-minimal hypersurfaces and self-shrinkers (2018). arXiv:1803.08268
Kapouleas, N., Kleene, S.J., Møller, N.M.: Mean curvature self-shrinkers of high genus: non-compact examples. J. Reine Angew. Math. 739, 1–39 (2018). https://doi.org/10.1515/crelle-2015-0050
Kleene, S., Møller, N.M.: Self-shrinkers with a rotational symmetry. Trans. Am. Math. Soc. 366(8), 3943–3963 (2014). https://doi.org/10.1090/S0002-9947-2014-05721-8
Liu, Z.H.: The Morse index of mean curvature flow self-shrinkers. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Massachusetts Institute of Technology (2016). https://dspace.mit.edu/handle/1721.1/104584
McGonagle, M.: Gaussian harmonic forms and two-dimensional self-shrinking surfaces. Proc. Am. Math. Soc. 143(8), 3603–3611 (2015). https://doi.org/10.1090/proc12750
McGrath, P.: Closed mean curvature self-shrinking surfaces of generalized rotational type (2015). arXiv:1507.00681
Møller, N.M.: Closed self-shrinking surfaces in \({\mathbb{R}}^3\) via the torus (2011). arXiv:1111.7318
Mramor, A.: Compactness and finiteness theorems for rotationally symmetric self shrinkers (2020). arXiv:2002.03465
Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. I. Trans. Am. Math. Soc. 361(4), 1683–1701 (2009). https://doi.org/10.1090/S0002-9947-08-04748-X
Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. II. Adv. Differ. Equ. 15(5–6), 503–530 (2010)
Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. Part III. Duke Math. J. 163(11), 2023–2056 (2014). https://doi.org/10.1215/00127094-2795108
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Berchenko-Kogan, Y. Bounds on the index of rotationally symmetric self-shrinking tori. Geom Dedicata 213, 83–106 (2021). https://doi.org/10.1007/s10711-020-00569-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-020-00569-9