A counterexample to the energy identity for sequences ofα-harmonic maps

Abstract

Abstract. We construct a closed Riemannian manifold (N, h) and a sequence of α-harmonic maps from S 2 into N with uniformly bounded energy such that the energy identity for this sequence is not true.Mathematics Subject Classification: 58E20, 35J60.

A gap theorem for $α$-harmonic maps between two-spheres

“…Note that these results rely crucially on the assumption that the target manifold is a round sphere since it is known that the energy identity is not true in general for a sequence of α-harmonic maps; see [10]. Finally, it follows from Theorem 2 in [3] that we have the decomposition…”

Morse Index Bound for Minimal Two Spheres

“…The α-energy approaches the usual energy as the parameter α in the perturbation goes to one, and the corresponding critical points of α-energy converges to a harmonic map. However, without curvature assumption [9] or finite fundamental group [4] for the ambient manifold (M, g), the harmonic spheres constructed by α-energy fails to realize the energy as α goes to one [8] [13, Remark 4.9.6]. Thus, we are motivated to prove the Morse index bound of the harmonic spheres produced by the min-max theory [1], which rules out the energy loss, namely: Theorem 1.1 (Main Theorem) Let (M, g) be a closed Riemannian manifold with dimension at least three, g generic and a nontrivial homotopy group π 3 (M).…”

Existence of min-max free boundary disks realizing the width of a manifold

“…However, we need another step in order to prove bubble convergence and energy identities : the "no-neck energy" lemma. Unfortunately, such a lemma is not true a priori for a sequence of α-harmonic maps, this was proved by Li-Wang [29]. Again, of course, we have more information than the α-harmonic map equation : the sequence comes from min-max solutions, and thanks to the monotonicity trick by Struwe [49], up to a subsequence, one can add an entropic condition that Lamm used [21] to prove the W 1,2 -bubble convergence.…”