On the Minimizers of Curvature Functionals in Asymptotically Flat Manifolds
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On the Minimizers of Curvature Functionals in Asymptotically Flat Manifolds Guodong Wei1 Received: 21 April 2020 / Accepted: 1 September 2020 © Mathematica Josephina, Inc. 2020
Abstract In this paper, we study the minimizers of curvature functionals (Willmore functional and extrinsic energy functional) subject to an area constraint in asymptotically flat manifolds. Under some certain conditions, we prove that such minimizers exist. Besides the surface theory related to the Willmore functional, the proofs also rely on the inverse mean curvature flow developed by Huisken and Ilmanen, on the positive mass theorem due to Schoen–Yau, and on the positive mass theorem for asymptotically flat manifolds with corners due to Miao and Shi–Tam. Our results may be of some interest in the General Relativity. Keywords Willmore functional · Extrinsic energy functional · Positive mass theorem · Inverse mean curvature flow Mathematics Subject Classification 58E30 · 53C20 · 53C24
1 Introduction Let (M, g) be a three-dimensional Riemannian manifold and be a closed surface in M. We denote by γ the induced metric on and we let dμ be its associated volume form. We also denote by H , A the associated mean curvature and second fundamental form, respectively. The Willmore functional of , denoted by W(), is defined by W() = |H |2 dμ.
Another important curvature functional of is the extrinsic energy functional E() = |A|2 dμ.
B 1
Guodong Wei [email protected] School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, People’s Republic of China
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G. Wei
Let R and Ric denote the scalar curvature and the Ricci curvature of (M, g), respectively. By the Gauss equation and the Gauss–Bonnet theorem, these two functionals have the following relation W() − E() = 4π χ () −
(R −2 Ric(ν, ν)) dμ,
(1.1)
where χ () is the Euler characteristic of , and ν is the unit outward normal vector along . It is easy to see from (1.1) that when (M, g) is R3 , W() and E() only differ by a topological constant. The significance of the above two functionals can be seen as follows. In some sense, the extrinsic energy functional can control the surface. Such as the curvature estimate of the minimal surface (cf. [6]), Hélein’s type convergence theorem (cf. [8,12,27]) and the − regularity theorem of the Willmore surface (cf. [22]). On the other hand, Willmore functional has been extensively studied since the early 20th century due in one hand to it’s rich mathematical signification but also to it’s importance in other area of science (such as in general relativity, mechanics, biology, optics). In general relativity, Willmore functional appears naturally in form of Hawking mass m H () of a surface which is used to measure the mass of the region enclosed by . In order to introduce the Hawking mass, we need to firstly recall the definition of the asymptotically flat manifold. Definition 1.1 A Riemannian 3-manifold (M 3 , g) is called asymptotically flat if there is a compact set K ⊂ M such that M\K is d
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