Stabilities of homothetically shrinking Yang-Mills solitons

Chen, Zhengxiang; Zhang, Yongbing

doi:10.1090/S0002-9947-2015-06467-8

Stabilities of homothetically shrinking Yang-Mills solitons
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by Zhengxiang Chen and Yongbing Zhang PDF

Trans. Amer. Math. Soc. 367 (2015), 5015-5041 Request permission

Abstract:

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and the second variation of the $\mathcal {F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity, and we obtain a gap theorem for homothetically shrinking solitons.

References

Ben Andrews, Haizhong Li, and Yong Wei, $\scr F$-stability for self-shrinking solutions to mean curvature flow, Asian J. Math. 18 (2014), no. 5, 757–777. MR 3287002, DOI 10.4310/AJM.2014.v18.n5.a1
Claudio Arezzo and Jun Sun, Self-shrinkers for the mean curvature flow in arbitrary codimension, Math. Z. 274 (2013), no. 3-4, 993–1027. MR 3078255, DOI 10.1007/s00209-012-1104-y
Jean-Pierre Bourguignon and H. Blaine Lawson Jr., Stability and isolation phenomena for Yang-Mills fields, Comm. Math. Phys. 79 (1981), no. 2, 189–230. MR 612248, DOI 10.1007/BF01942061
H.-D. Cao, R. S. Hamilton, and T. Ilmanen, Gaussian density and stability for some Ricci solitons, arXiv:math/0404165
Huai-Dong Cao and Haizhong Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), no. 3-4, 879–889. MR 3018176, DOI 10.1007/s00526-012-0508-1
Huai-Dong Cao and Meng Zhu, On second variation of Perelman’s Ricci shrinker entropy, Math. Ann. 353 (2012), no. 3, 747–763. MR 2923948, DOI 10.1007/s00208-011-0701-0
Chen Yunmei and Shen Chun-Li, Monotonicity formula and small action regularity for Yang-Mills flows in higher dimensions, Calc. Var. Partial Differential Equations 2 (1994), no. 4, 389–403. MR 1383915, DOI 10.1007/BF01192090
Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755–833. MR 2993752, DOI 10.4007/annals.2012.175.2.7
Paul M. N. Feehan, Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds, arXiv:1409.1525.
Andreas Gastel, Singularities of first kind in the harmonic map and Yang-Mills heat flows, Math. Z. 242 (2002), no. 1, 47–62. MR 1985449, DOI 10.1007/s002090100306
Joseph F. Grotowski, Finite time blow-up for the Yang-Mills heat flow in higher dimensions, Math. Z. 237 (2001), no. 2, 321–333. MR 1838314, DOI 10.1007/PL00004871
Richard S. Hamilton, Monotonicity formulas for parabolic flows on manifolds, Comm. Anal. Geom. 1 (1993), no. 1, 127–137. MR 1230277, DOI 10.4310/CAG.1993.v1.n1.a7
Min-Chun Hong and Gang Tian, Global existence of the $m$-equivariant Yang-Mills flow in four dimensional spaces, Comm. Anal. Geom. 12 (2004), no. 1-2, 183–211. MR 2074876, DOI 10.4310/CAG.2004.v12.n1.a10
Nam Q. Le and Natasa Sesum, Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers, Comm. Anal. Geom. 19 (2011), no. 4, 633–659. MR 2880211, DOI 10.4310/CAG.2011.v19.n4.a1
Yng-Ing Lee and Yang-Kai Lue, The stability of self-shrinkers of mean curvature flow in higher co-dimension, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2411–2435. MR 3301868, DOI 10.1090/S0002-9947-2014-05969-2
Jiayu Li and Yongbing Zhang, Lagrangian F-stability of closed Lagrangian self-shrinkers, Preprint.
Casey Kelleher and Jeff Streets, Entropy, stability, and Yang-Mills flow, arXiv: 1410.4547.
Hisashi Naito, Finite time blowing-up for the Yang-Mills gradient flow in higher dimensions, Hokkaido Math. J. 23 (1994), no. 3, 451–464. MR 1299637, DOI 10.14492/hokmj/1381413099
Johan Råde, On the Yang-Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992), 123–163. MR 1179335, DOI 10.1515/crll.1992.431.123
Andreas E. Schlatter, Michael Struwe, and A. Shadi Tahvildar-Zadeh, Global existence of the equivariant Yang-Mills heat flow in four space dimensions, Amer. J. Math. 120 (1998), no. 1, 117–128. MR 1600272, DOI 10.1353/ajm.1998.0004
Michael Struwe, The Yang-Mills flow in four dimensions, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 123–150. MR 1385523, DOI 10.1007/BF01191339
Ben Weinkove, Singularity formation in the Yang-Mills flow, Calc. Var. Partial Differential Equations 19 (2004), no. 2, 211–220. MR 2034580, DOI 10.1007/s00526-003-0217-x
Yongbing Zhang, $\scr {F}$-stability of self-similar solutions to harmonic map heat flow, Calc. Var. Partial Differential Equations 45 (2012), no. 3-4, 347–366. MR 2984136, DOI 10.1007/s00526-011-0461-4