Foliations of asymptotically flat manifolds by surfaces of Willmore type

Lamm, Tobias; Metzger, Jan; Schulze, Felix

doi:10.1007/s00208-010-0550-2

Cited by 49 publications

(82 citation statements)

References 19 publications

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“…Such manifolds were called asymptotically Schwarzschild in [LMS11] and we will adopt this terminology from now on. The Schwarzschild manifold (M S , g S ) = (R 3 \ {0}, g S ) is the model space for the problem studied in this paper.…”

Section: Preliminariesmentioning

confidence: 99%

“…Hence, the problem of maximizing the Hawking mass even with fixed area seems particularly challenging from a variational point of view. In this work we make partial progress in understanding the role of the centred spheres in the Schwarzschild space or more generally of the leaves Σ λ in the foliation constructed in [LMS11] in asymptotically Schwarzschild spaces regarding the maximization of the Hawking mass. In fact, we show the following: Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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

Koerber

2020

J Geom Anal

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We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C 3 −close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type, see [LMS11]. In this paper, we prove that the leaves of this foliation are stable under small area preserving W 2,2 −perturbations with respect to the area preserving Willmore flow. This implies, in particular, that the leaves are strict local area preserving maximizers of the Hawking mass with respect to the W 2,2 −topology.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Koerber

2020

J Geom Anal

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“…We have then the following formulas, see Section 3 in . Proposition With the above notation we have the formulas

W^{'} false(i (Σ) false) false[φ false] = \int_{normalΣ} (L H + \frac{1}{2} H^{3}) φ d σ

and

\begin{matrix} true \\ right W^{''} (i false(normalΣ false)) [φ, φ] & = & left 2 \int_{Σ} ([] {(L φ)}^{2} + \frac{1}{2} H^{2} {| \nabla φ |}^{2} - 2 \overset{˚}{A} false(\nabla φ, \nabla φ false)) d σ \\ left + 2 \int_{Σ} φ^{2} {false(false| \nabla H false|}_{g}^{2} + 2 ϖ (\nabla H) + H \nabla H + 2 g (\nabla^{2} H, \overset{˚}{A}) + 2 H^{2} {false| \overset{A}{true} false|}_{g}^{2} \\ left + 2 H g (\overset{˚}{A}, T) - H g (\nabla_{n} Ric) (n, n) - \frac{1}{2} H^{2} {false| A false|}_{g}^{2} - \frac{1}{2} H^{2} Ric (n, n) false) d σ \\ left + \int_{Σ} (() L H + \frac{1}{2} H^{3}) ((), \frac{\partial φ}{\partial}) \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…Let us recall that area‐constrained Willmore surfaces satisfy the equation

{normalΔ}_{\overset{g}{true}} H + H {| \overset{˚}{A} |}^{2} + H Ric false(n, n false) = λ H,

for some

λ \in R

playing the role of Lagrange multiplier. These immersions are naturally linked to the Hawking mass

m_{H} false(i false) : = \frac{\sqrt{A r e a false(i false)}}{64 π^{3 / 2}} (16 π - W false(i false)),

since, clearly, the critical points of the Hawking mass under area constraint are exactly the area‐constrained Willmore immersions (see and the references therein for more material about the Hawking mass).…”

Section: Introductionmentioning

confidence: 99%

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

Ikoma

Malchiodi

Mondino

2017

Proc. London Math. Soc.

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We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behavior of degenerating tori under the action of the Möbius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.

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“…In more generality, it is interesting to study the minimization problems associated to integral functionals depending on the curvatures of the type introduced by Willmore (see [33]) and studied in the euclidean space (see for instance the works of Simon [27], Kuwert and Schätzle [12], Rivière [30]) or in Riemannian manifolds (see, for example, [13], [21] and [22]). …”

Section: Introductionmentioning

confidence: 99%

Existence of integral $$m$$ -varifolds minimizing $$\int \!|A|^p$$ and $$\int \!|H|^p,\,p>m,$$ in Riemannian manifolds

Mondino

2012

Calc. Var.

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abstract. We prove existence of integral rectifiable m-dimensional varifolds minimizing functionals of the type |H| p and |A| p in a given Riemannian n-dimensional manifold (N, g), 2 ≤ m < n and p > m, under suitable assumptions on N (in the end of the paper we give many examples of such ambient manifolds). To this aim we introduce the following new tools: some monotonicity formulas for varifolds in R S involving |H| p , to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals.

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Foliations of asymptotically flat manifolds by surfaces of Willmore type

Cited by 49 publications

References 19 publications

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

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

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

Existence of integral $$m$$ -varifolds minimizing $$\int \!|A|^p$$ and $$\int \!|H|^p,\,p>m,$$ in Riemannian manifolds

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