Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

Huang, Tao; Liu, Lei; Luo, Yong; Wang, Changyou

doi:10.1007/bf03377389

Cited by 4 publications

(2 citation statements)

References 26 publications

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“…The corresponding heat flows have been studied extensively as well, as a tool to prove the existence of biharmonic maps and polyharmonic maps in a given homotopy class, see e.g. [26,27,13,45,20,21,32]. It should be noted that the extrinsic and intrinsic cases come in two different flavors: the intrinsic variants are considered more geometrically natural because they do not depend on the embedding of the target manifold N ֒→ R K , although they are less natural from the variational point of view due to the lack of coercivity for the intrinsic energies (and thus they are considerably more difficult analytically and less studied); on the other hand, the extrinsic variants are more natural from the analytical point of view but in turn they do depend on the embedding of N ֒→ R K .…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Lin¹,

Zhu²

2022

Preprint

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Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds M and N were introduced in search of a natural notion of "harmonicity" for maps defined on a general even dimensional Riemannian manifold M. They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions.In this paper, we consider conformal-harmonic maps from certain locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.

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Section: Introductionmentioning

confidence: 99%

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Lin¹,

Zhu²

2022

Preprint

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“…where u 0 ∈ W 2,2 (B 1 , N ). There are several results of existence for extrinsic bi-harmonic map heat flow, see for instance Lamm [24] for small initial data and Gastel [11] and Wang [41] for solutions with many finitely singular time and any initial data, see also [17] and [18]. Moreover, the solutions to (8) satisfy the following energy inequality…”

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confidence: 99%

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

Laurain,

Lin

2018

Preprint

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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].

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Doubling annulus Pohožaev type identity and applications to approximate biharmonic maps

Chen,

Zhu

2023

Calc. Var.

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Heat Flow of Extrinsic Biharmonic Maps from a four Dimensional Manifold with Boundary

Cited by 4 publications

References 26 publications

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

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

Doubling annulus Pohožaev type identity and applications to approximate biharmonic maps

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