Energy convexity of intrinsic bi-harmonic maps and applications I: spherical target

Abstract

Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of coercivity for the intrinsic bi-energy. In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B 1 ⊂ R 4 into the sphere S n . This is a higher-order analogue of the energy convexity and uniqueness for weakly harmonic maps on unit 2-disk in R 2 proved by Colding and Minicozzi [8] (see also Lamm and the second author [26]). In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. As an application, we also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into S n , which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [25]. Moreover, the energy convexity along the flow yields the uniform convergence of the flow which is not known before. One of the key ingredients in our proofs is a refined version of the ǫ-regularity of the first author and Rivière [28].

Min-max minimal disks with free boundary in Riemannian manifolds

“…The key geometric observation in the proof of the energy convexity lies in two orthogonality conditions, one observed by Colding-Minicozzi [6], and the other by the third author [56]. Very recently, this idea also permits the first author and Laurain to obtain an energy convexity and uniqueness for weakly intrinsic bi-harmonic maps defined on the unit 4-ball with small bi-energy, which in particular yields a version of uniqueness for weakly harmonic maps in dimension 4, see [27]. The other ingredient is the uniform continuity estimate up to the free boundary for weakly harmonic maps.…”

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

“…Besides the uniqueness consequence, this theorem is very useful for many other problems such as flow convergence, see [25]. In fact our proof simplifies the original proof by Colding-Minicozzi and the one of Lamm-Lin [22].…”

“…In fact, the crucial point is the compactness compensation phenomena which appears in all conformally invariant problems. This remark permits the first author and Lin [25] to prove the convexity for some biharmonic maps, see also [24] for a review. In the present paper we prove the following free boundary version of Colding-Minicozzi energy convexity.…”

“…This kind of energy convexity is also of first importance to get some strong convergence of the flow associated to some conformally invariant problem such as harmonic maps or bi-harmonic maps, see [30] and [25]. Moreover, using the flow as another smoother for sweepouts could lead to some compactness, as in Fraser-Schoen [12].…”