The Area Preserving Willmore Flow and Local Maximizers of the Hawking Mass in Asymptotically Schwarzschild Manifolds

Koerber, Thomas

doi:10.1007/s12220-020-00401-6

Cited by 7 publications

(11 citation statements)

References 31 publications

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“…Obviously we have λ(Σ) ≤ maxx∈Σ |x|. Using the terminology in [6], every admissible surface Σ satisfying (12) with R, δ −1 ≫ 1 is on-center For the class of admissible surfaces, we have the following well-posedness result for (1): 17,Thm. 5.3]).…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Stationary solution to (1) are called surfaces of Willmore type. The existence and stability of such surfaces are studied in [6,17].…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Remark 2. The map Φ in Theorem 3 is defined by Φ(r, z) = θ(Φ(r, z), r, z), where θ, Φ are given respectively in (17) and Definition 4. The content of Theorem 3 says that the static and dynamical properties of ( 5) are captured by the effective action W • Φ, on the 3-manifold given by the level set…”

Section: Main Results and Organization Of The Papermentioning

confidence: 99%

“…To use results in [6,17,18], we assume the scalar curvature Sc on M satisfies the following decay properties:…”

Section: Backgroundsmentioning

confidence: 99%

“…We say a point ζ * ∈ M ′ is the barycenter of x * if ζ * solves the following algebraic system: Σ . See [17] and the references therein. Our version of barycenter retains the key decay property as [17,Sec.…”

Section: Barycentermentioning

confidence: 99%

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Adiabatic theory for the area-constrained Willmore flow

Zhang

2021

Preprint

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In this paper, we develop an adiabatic theory for the evolution of large closed surfaces under the area-constrained Willmore (ACW) flow in a three-dimensional asymptotically Schwarzschild manifold. We construct explicitly a map, defined on a certain four-dimensional manifold of barycenters, which characterizes key static and dynamical properties of the ACW flow.In particular, using this map, we find an explicit four-dimensional effective dynamics of barycenters, which serves as a uniform asymptotic approximation for the (infinite-dimensional) ACW flow, so long as the initial surface satisfies certain mild geometric constraints (which determine the validity interval).Conversely, given any prescribed flow of barycenters evolving according this effective dynamics, we construct a family of surfaces evolving by the ACW flow, whose barycenters are uniformly close to the prescribed ones on a large time interval (whose size depends on the geometric constraints of initial configurations).

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Section: Preliminary Resultsmentioning

confidence: 99%

“…Stationary solution to (1) are called surfaces of Willmore type. The existence and stability of such surfaces are studied in [6,17].…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Main Results and Organization Of The Papermentioning

confidence: 99%

“…To use results in [6,17,18], we assume the scalar curvature Sc on M satisfies the following decay properties:…”

Section: Backgroundsmentioning

confidence: 99%

Section: Barycentermentioning

confidence: 99%

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Adiabatic theory for the area-constrained Willmore flow

Zhang

2021

Preprint

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The Willmore Center of Mass of Initial Data Sets

Eichmair

Koerber

2022

Commun. Math. Phys.

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We refine the Lyapunov–Schmidt analysis from our recent paper (Eichmair and Koerber in Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds. arXiv preprint arXiv:2101.12665, 2021) to study the geometric center of mass of the asymptotic foliation by area-constrained Willmore surfaces of initial data for the Einstein field equations. If the scalar curvature of the initial data vanishes at infinity, we show that this geometric center of mass agrees with the Hamiltonian center of mass. By contrast, we show that the positioning of large area-constrained Willmore surfaces is sensitive to the distribution of the energy density. In particular, the geometric center of mass may differ from the Hamiltonian center of mass if the scalar curvature does not satisfy additional asymptotic symmetry assumptions.

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Refinement of Hélein’s conjecture on boundedness of conformal factors when $$n = 3$$

Плотников

Toland

2023

Annali di Matematica

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For smooth mappings of the unit disc into the oriented Grassmannian manifold $${\mathbb {G}}_{n,2}$$ G n , 2 , Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $$|{{\varvec{A}}} |^2$$ | A | 2 , the squared-length of the second fundamental form, is less than $$\gamma _n=8\pi $$ γ n = 8 π . It has since been shown that the optimal bounds that guarantee this result are: $$\gamma _3 = 8\pi $$ γ 3 = 8 π and $$\gamma _n = 4\pi $$ γ n = 4 π for $$n \ge 4$$ n ≥ 4 . For isothermal immersions in $${\mathbb {R}}^3$$ R 3 , this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than $$\gamma _3$$ γ 3 . The goal here is to prove that when $$n=3$$ n = 3 the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when $$|{{\varvec{A}}} |^2$$ | A | 2 is integrable and the integral of $$|K |$$ | K | , where K is the Gauss curvature, is less than $$4\pi $$ 4 π . Since $$2|K |\le |{{\varvec{A}}} |^2$$ 2 | K | ≤ | A | 2 this implies the known result for isothermal immersions, but $$|K |$$ | K | may be small when $$|{{\varvec{A}}} |^2$$ | A | 2 is large. The method, which is purely analytic, is then developed to examine the case $$n=3$$ n = 3 when $$|\varvec{A} |$$ | A | is only square-integrable. The possibility of extending that result in the language of Grassmannian manifolds to the case $$n>3$$ n > 3 is outlined in an Appendix.

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The Area Preserving Willmore Flow and Local Maximizers of the Hawking Mass in Asymptotically Schwarzschild Manifolds

Cited by 7 publications

References 31 publications

Adiabatic theory for the area-constrained Willmore flow

Adiabatic theory for the area-constrained Willmore flow

The Willmore Center of Mass of Initial Data Sets

Refinement of Hélein’s conjecture on boundedness of conformal factors when $$n = 3$$

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