Stabilities of homothetically shrinking Yang-Mills solitons

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 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.

Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case

“…Hence, the integral in (3.8) can be explicitly computed, and this yields B(8, 4) < 1. In the same way we prove that the same holds for B(9, 4), B(10, 6) and B (11,6).…”

“…Related results. For the Yang-Mills heat flow, notions of variational stability of shrinking solitons have been investigated by Kelleher and Streets [22] as well as by Chen and Zhang [6]. However, to the best of our knowledge, our results together with [11] are the only ones that prove stable blowup behavior in any sense.…”

Stable blowup for the supercritical Yang–Mills heat flow

“…These objects correspond to solutions of the YM heat flow on the trivial bundle over R d , which is our main motivation to study the problem in this geometrical setting. Moreover, Weinkove [34] as well as by Chen and Zhang [9]. However, to the best of our knowledge no rigorous proof on the stability of the Weinkove solution given in Eq.…”

“…The results of Weinkove have raised interest in the stability of YM-solitons in recent years and notions of variational stability have been introduced by Kelleher and Streets [34] as well as by Chen and Zhang [9]. However, to the best of our knowledge no rigorous proof on the stability of the Weinkove solution given in Eq.…”

Lagrangian F-stability of closed Lagrangian self-shrinkers

“…Colding and Minicozzi's idea of classifying self-similar solutions by employing entropystability and F-stability also applies to other geometric flows, for the harmonic map heat flow case see [42] and for the Yang-Mills flow case see [13]. For Ricci shrinkers and Ricciflat manifolds, an analogous stability to the F-stability is the linear stability.…”