Abstract
We revisit the interplay between the mass, the center of mass and the large scale behavior of certain isoperimetric quotients in the setting of asymptotically flat 3-manifolds (both without and with a non-compact boundary). In the boundaryless case, we first check that the isoperimetric deficits involving the total mean curvature recover the ADM mass in the asymptotic limit, thus extending a classical result due to G. Huisken. Next, under a Schwarzschild asymptotics and assuming that the mass is positive we indicate how the implicit function method pioneered by R. Ye and refined by L.-H. Huang may be adapted to establish the existence of a foliation of a neighborhood of infinity satisfying the corresponding curvature conditions. Recovering the mass as the asymptotic limit of the corresponding relative isoperimetric deficit also holds true in the presence of a non-compact boundary, where we additionally obtain, again under a Schwarzschild asymptotics, a foliation at infinity by free boundary constant mean curvature hemispheres, which are shown to be the unique relative isoperimetric surfaces for all sufficiently large enclosed volume, thus extending to this setting a celebrated result by M. Eichmair and J. Metzger. Also, in each case treated here we relate the geometric center of the foliation to the center of mass of the manifold as defined by Hamiltonian methods.
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References
Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)
Lee, J.M., Parker, T.: The Yamabe problem. Bull. AMS 17(1), 37–91 (1987)
Brendle, S., Marques, F.: Recent progress on the Yamabe problem. Adv. Lect. Math. 20, 29–47 (2011)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)
Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996)
Ye, R.: Foliation by constant mean curvature spheres on asymptotically flat manifolds. In: Geometric Analysis and the Calculus of Variations (1996)
Metzger, J.: Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. J. Differ. Geom. 77(2), 201–236 (2007)
Huang, L.-H.: On the center of mass of isolated systems with general asymptotics. Class. Quantum Gravity 26(1), 015012 (2008)
Huang, L.-H.: Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics. Commun. Math. Phys. 300(2), 331–373 (2010)
Nerz, C.: Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry. J. Differ. Geom. 109(2), 257–289 (2018)
Eichmair, M., Metzger, J.: Large isoperimetric surfaces in initial data sets. J. Differ. Geom. 94(1), 159–186 (2013)
Regge, T., Teitelboim, C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. 88(1), 286–318 (1974)
Beig, R., Ó Murchadha, N.: The Poincaré group as the symmetry group of canonical general relativity. Ann. Phys. 174(2), 463–498 (1987)
Almaraz, S.: Convergence of scalar-flat metrics on manifolds with boundary under a Yamabe-type flow. J. Differ. Equ. 259(7), 2626–2694 (2015)
Almaraz, S., de Queiroz, O.S., Wang, S.: A compactness theorem for scalar-flat metrics on 3-manifolds with boundary. J. Funct. Anal. 277(7), 2092–2116 (2019)
Almaraz, S., Barbosa, E., de Lima, L.L.: A positive mass theorem for asymptotically flat manifolds with a non-compact boundary. Commun. Anal. Geom. 24(4), 673–715 (2016)
Almaraz, S., de Lima, L.L., Mari, L.: Spacetime positive mass theorems for initial data sets with noncompact boundary. In: International Mathematics Research Notices (2020)
de Lima, L.L., Girão, F., Montalbán, A.: The mass in terms of Einstein and Newton. Class. Quantum Gravity 36(7), 075017 (2019)
Huisken, G.: An isoperimetric concept for mass and quasilocal mass. Oberwolfach Rep 3(1), 87–88 (2006)
Huang, L.-H.: Center of mass and constant mean curvature foliations for isolated systems. MSRI Lecture Notes (2009)
Schneider, R.: Convex bodies: the Brunn–Minkowski theory. In: Encyclopedia of Mathematics and its Applications, vol. 151. Cambridge University Press (2014)
Guan, P., Li, J.: The quermassintegral inequalities for \(k\)-convex starshaped domains. Adv. Math. 221(5), 1725–1732 (2009)
Chang, S.-Y.A., Wang, Y.: On Aleksandrov–Fenchel inequalities for k-convex domains. Milan J. Math. 79(1), 13–38 (2011)
Fall, M.M.: Area-minimizing regions with small volume in Riemannian manifolds with boundary. Pac. J. Math. 244(2), 235–260 (2009)
Montenegro, J.F.: Foliation by free boundary constant mean curvature leaves. ar**v:1904.11867 (2019)
Eichmair, M., Metzger, J.: Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions. Invent. Math. 194(3), 591–630 (2013)
Munoz Flores, A.E., Nardulli, S.: The isoperimetric problem of a complete Riemannian manifold with a finite number of asymptotically Schwarzschild ends. Commun. Anal. Geom. 28(7), 1577–1601 (2020)
Nerz, C.: Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry. Calc. Var. Partial Differ. Equ. 54(2), 1911–1946 (2015)
Chodosh, O., Eichmair, M., Shi, Y., Yu, H.: Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian 3-manifolds. Commun. Pure Appl. Math. 74(4), 865–905 (2021)
Jauregui, J.L., Lee, D.A.: Lower semicontinuity of mass under \({C}^0\) convergence and Huisken’s isoperimetric mass. Journal für die reine und angewandte Mathematik 2019(756), 227–257 (2019)
Cederbaum, C., Sakovich, A.: On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in general relativity. Calc. Var. Partial Differ. Equ. 60(6), 1–57 (2021)
Chen, P.-N., Wang, M.-T., Yau, S.-T.: Quasilocal angular momentum and center of mass in general relativity. Adv. Theor. Math. Phys. 20, 671–682 (2016)
Gray, A.: Tubes. Progress in Mathematics, vol. 221. Birkhäuser, Basel (2012)
Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Gravitation: An Introduction to Current Research (1962)
Christodoulou, D.: Mathematical Problems of General Relativity I, vol. 1. European Mathematical Society, Helsinki (2008)
Harlow, D., Wu, J.: Covariant phase space with boundaries. J. High Energy Phys. 2020(10), 1–52 (2020)
de Lima, L.L.: Conserved quantities in general relativity: the case of initial data sets with a non-compact boundary. To appear in “Perspectives in Scalar Curvature”, edited by M. Gromov and H.B. Lawson (2022)
Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)
Fan, X.-Q., Shi, Y., Tam, L.-F.: Large-sphere and small-sphere limits of the Brown–York mass. Comm. Anal. Geom. 17(2), 37–72 (2009)
Lee, D.A., LeFloch, P.G.: The positive mass theorem for manifolds with distributional curvature. Commun. Math. Phys. 339(1), 99–120 (2015)
Michel, B.: Geometric invariance of mass-like asymptotic invariants. J. Math. Phys. 52(5), 052504 (2011)
Chan, P.-Y., Tam, L.-F.: A note on center of mass. Commun. Anal. Geom. 24(3), 471–486 (2016)
Cederbaum, C., Nerz, C.: Explicit Riemannian manifolds with unexpectedly behaving center of mass. Ann. Henri Poincaré 16(7), 1609–1631 (2015)
Qing, J., Tian, G.: On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds. J. Am. Math. Soc. 20(4), 1091–1110 (2007)
Corvino, J., Wu, H.: On the center of mass of isolated systems. Class. Quantum Gravity 25(8), 085008 (2008)
de Lima, L.L., Lázaro, I.C.: A Cauchy–Crofton formula and monotonicity inequalities for the Barbosa–Colares functionals. Asian J. Math. 7(1), 81–89 (2003)
Reilly, R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26(3), 459–472 (1977)
Máximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex three-manifolds. J. Differ. Geom. 106(1), 139–186 (2017)
Bray, H.L.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Ph.D. Thesis, Stanford University (1997)
Corvino, J., Gerek, A., Greenberg, M., Krummel, B.: On isoperimetric surfaces in general relativity. Pac. J. Math. 231(1), 63–84 (2007)
Ritoré, M., Rosales, C.: Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones. Trans. Am. Math. Soc. 356(11), 4601–4622 (2004)
Almaraz, S., de Lima, L.L.: The mass of an asymptotically hyperbolic manifold with a non-compact boundary. Ann. Henri Poincaré 21(11), 3727–3756 (2020)
Rigger, R.: The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscr. Math. 113(4), 403–421 (2004)
Neves, A., Tian, G.: Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19(3), 910 (2009)
Neves, A., Tian, G.: Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds II. J. für die reine und angewandte Mathematik 2010(641), 69–93 (2010)
Mazzeo, R., Pacard, F.: Constant curvature foliations in asymptotically hyperbolic spaces. Revista Matematica Iberoamericana 27(1), 303–333 (2011)
Chodosh, O.: Large isoperimetric regions in asymptotically hyperbolic manifolds. Commun. Math. Phys. 343(2), 393–443 (2016)
Ros, A., Vergasta, E.: Stability for hypersurfaces of constant mean curvature with free boundary. Geom. Dedicata. 56(1), 19–33 (1995)
Ros, A., Souam, R.: On stability of capillary surfaces in a ball. Pac. J. Math. 178(2), 345–361 (1997)
Barbosa, J.L.M., Colares, A.G.: Stability of hypersurfaces with constant \(r\)-mean curvature. Ann. Glob. Anal. Geom. 15(3), 277–297 (1997)
Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bulletin des Sciences Mathématiques 117(2), 211–239 (1993)
Alías, L.J., Brasil, A., Colares, A.G.: Integral formulae for spacelike hypersurfaces in conformally stationary spacetimes and applications. Proc. Edinb. Math. Soc. 46(2), 465–488 (2003)
Chai, X.: Two quasi-local masses evaluated on surfaces with boundary. ar**v preprint ar**v:1811.06168 (2018)
Ambrozio, L.C.: Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds. J. Geom. Anal. 25(2), 1001–1017 (2015)
Acknowledgements
S. Almaraz has been partially supported by CNPq/Brazil Grant 309007/2016-0 and FAPERJ/Brazil Grant 202.802/2019, and L. de Lima has been partially supported by CNPq/Brazil grant 312485/2018-2. Both authors have been partially supported by FUNCAP/CNPq/PRONEX Grant 00068.01.00/15. Also, the authors thank A. Freitas, E. Lima, J.F. Montenegro and S. Nardulli for conversations at an early stage of this project.
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Appendices
Appendix A: The large scale isoperimetric deficits and the mass: the proofs of Theorems 4 and 7
The arguments to prove Theorems 4 and 7 are simple variations on the computation appearing in [39, Section 2], where a proof of Theorem 2 appears. This justifies the inclusion of a somewhat detailed account of their calculation in what follows.
If (M, g) is asymptotically flat as in Definition 1, we first observe that, since \(\partial r/\partial x_i=x_i/r\), we have
and hence
Also, if \(\nu \) is the outward unit normal vector field to the coordinate 2-sphere \(S^2_r\) then
Let \(dS^{2,\delta }_r=r^2dS^{2,\delta }_1\) be the area element of the Euclidean sphere of radius r. It follows that the area element of the corresponding coordinate sphere \(S^2_r\) expands as
where
is the induced metric (extended to vanish in the radial direction). Thus, the area of \(S^{2}_r\) is
From this we obtain
where the comma means partial differentiation. Using (A5) we get
We now work out the third term in the right-hand side. We first note that
We then compute:
so we have
Using (A5) we have
Also, integration by parts together with (A7) gives
so that
Thus,
Combining this with (A6), we get
We now look at the volume V(r) enclosed by \(S_r^2\). By the co-area formula, (A2) and (A4),
We may now eliminate the integral term in (A10) and (A11). The result is
Integrating we obtain a formula relating the volume and area, namely,
which gives
On the other hand, from (A6),
so that
which gives the proof of Theorem 2.
So far we have been following [39] closely. We now explain how a little variation yields the proof of Theorem 4. We will make use of the well-known expansion
Together with (A4) this gives
Also, by the first variation formula for the area,
and combining this with (A10) we get
We now use (A12) to eliminate the area term. Solving for the volume we get
so that, using (A14),
which finishes the proof of the first equality in (11). As for the second one, note that by (A15) and (A14),
which completes the proof of Theorem 4.
We now present the proof of Theorem 7. We first observe that instead of (A6) we now have
Also, the integration by parts leading to (A8) now produces an extra term, so that (II) gets replaced by
Thus, instead of (A10) we now have
Hence, proceeding exactly as before we now get
which gives
where
This completes the proof of Theorem 7.
Appendix B: The variational setup
Here we address the variational issues needed in the bulk of the paper. Our aim is twofold. First, we review the well-known variational theory of free boundary constant mean curvature surfaces [58, 59] in a way that is convenient for our purposes. Next, we discuss the much less known variational theory of closed surfaces which are critical for the total mean curvature functional under a volume preserving constraint and develop the corresponding stability theory. We remark that the variational theory associated to curvature integrals involving elementary symmetric functions of the principal curvatures (quermassintegrals) of hypersurfaces in space forms is a well established subject; see [60] and the references therein.
We start by considering a one-parameter family of compact, embedded surfaces \(t\in (-\varepsilon ,\varepsilon )\mapsto S_t\) in an arbitrary Riemannian manifold \((M^3,g)\) evolving as
where \(x_t\) is the smooth map defining the embedding and \(Y_t\) is a vector field along \(S_t\), a (not necessarily normal) section of TM restricted to \(S_t\). As usual, if \(\nu _t\) is the unit normal vector field along \(S_t\), let \(W=\nabla \nu _t\) be the shape operator of \(S_t\), so the corresponding principal curvatures (the eigenvalues of W) are \(\kappa _1\) and \(\kappa _2\). Thus, the mean curvature is \(H=\kappa _1+\kappa _2\) and the Gauss–Kronecker curvature is \(K=\kappa _1\kappa _2\). For later reference, we recall that
is the modified Gauss–Kronecker curvature.
A well-known computation gives
where A(t) is the area of \(S_t\) and we agree to drop the subscript t upon evaluation at \(t=0\). Next we decompose \(Y_t\) into its normal and tangential components:
Thus, if we assume further that \(S_t\) carries a boundary \(\partial S_t\),
where \(\mu \) is the outward unit normal vector field along \(\partial S\) and we used that \(H=\textrm{div}_S\nu \).
Let us assume now that M also carries a boundary, say \(\Sigma \), with the variation being admissible in the sense that \(\partial S_t\subset \Sigma \). It follows that \(S=S_0\) is critical for the area under such variations satisfying the volume preserving condition
if and only if the mean curvature is constant and S meets \(\Sigma \) orthogonally along \(\partial S\). We then say that S is a free boundary constant mean curvature (CMC) surface.
We now recall the corresponding notion of stability. Assuming that \(S=S_0\) is a free boundary CMC as above, a well-known computation [59] gives the second variational formula for the area:
where
\(\kappa =\langle \nu ,{\mathcal {W}} \nu \rangle \) and \({\mathcal {W}}=\nabla \eta \) is the shape operator of the embedding \(\Sigma \hookrightarrow M\). Here, \(\eta \) is the outward unit normal vector to M along \(\Sigma \).
Recall that \(S=S_0\) is strictly stable (as a free boundary CMC surface) if the right-hand side of (B6) is positive for any \(f\ne 0\) satisfying (B5). Accordingly, we define
Proposition 23
A free boundary CMC surface S as above is strictly stable if and only if the first eigenvalue \(\lambda _{{\mathscr {L}}_S}\) of the eigenvalue problem
is positive, where \(f\in {\mathcal {F}} (S)\). Equivalently, for any \(0\ne f\in {\mathcal {F}}(S)\),
We now turn to the variational theory of the total mean curvature functional \(\int _S H dS\). Here we assume that \(S_t\) is closed (\(\partial S=\emptyset \)) and the variation is normal (\(Y=f\nu \)). A simple computation shows that the shape operator evolves as
where \(\nabla _S^2\), the Hessian of f, is viewed as a (1, 1)-tensor, \(\textrm{Riem}^\nu _g(\cdot )=\textrm{Riem}_g(\cdot ,\nu )\nu \) and \({{W^2=W\circ W}}\).
Proposition 24
In a Riemannian 3-manifold (M, g) as above, a closed surface extremizes the total mean curvature under volume (respectively, area) preserving variations if and only if \({{\widetilde{K}}}=\textrm{const}\) (respectively, \({{\widetilde{K}}}=\gamma H\), where \(\gamma \) is a constant).
Proof
From \(\partial dS_t/\partial t=fHdS_t\), the fact that the mean curvature evolves as
and the algebraic identity \(\vert W\vert ^2=H^2-2K\), we immediately see that
which proves the first statement. As for the second one, just combine the computation above with (B4) and take into account that \(\partial S=\emptyset \). \(\square \)
In order to discuss the stability of this variational problem, we now compute the variation of \({{\widetilde{K}}}\). First, from \(\partial \nu /\partial t=-\nabla _Sf\),
As for the variation of K, we first recall the well-known formula
where \(\Pi =HI-W\) is the Newton tensor [61]. Using (B8) we then get
To proceed we choose an orthonormal frame \(e_A\), \(A=1,2\), tangent to S with \((\nabla _S)_{e_A}e_B=0\) at the given point. We compute
where the semicolon denotes covariant derivation. By Codazzi equations, recalling that \(h=g\vert _S\),
so that
Thus, from (B11) and the algebraic identity \(\textrm{tr}_S(\Pi W^2)=HK\), which holds in dimension 3,
where
Together with (B10) this finally gives
where
is the corresponding Jacobi operator. We note that
for any functions f and \({{\tilde{f}}}\). In particular, \(L_S\) is always self-adjoint. Moreover, it is easy to check that this operator is elliptic whenever \(\Pi \) is positive definite.
We now consider a surface \(S\subset M\) satisfying \({{\widetilde{K}}}=\mathrm{const.}\) and with the property that \(\Pi \) is positive definite everywhere. We then say that S is strictly stable if
for any normal variation as in (B1) with \(f\ne 0\). As before let us set
Proposition 25
S is strictly stable if and only if
for any \(0\ne f\in {\mathcal {G}}(S)\). Equivalently, the first eigenvalue \(\lambda _{L_S}\) of the eigenvalue problem
is positive.
Appendix C: The Gauss–Kronecker curvature in terms of the mean curvature
In this section we prove Proposition 13. Thus we aim to prove the identity (33) which expresses the Gauss–Kronecker curvature in terms of the mean curvature up to terms decaying fast enough at infinity. Our starting point is the fact that the radial vector field
is conformal with respect to the Euclidean metric, i.e., \(\mathcal L_X\delta =2\delta \), where \({\mathcal {L}}\) is the Lie derivative. From this we see that X is also conformal with respect to the metric \(f_{m,c}^{\gamma _1, \gamma _2}\delta \) where
for some \(\gamma _1, \gamma _2\in {\mathbb {R}}\) and \(c\in {\mathbb {R}}^3\). Indeed, there holds \({\mathcal {L}}_X(f_{m,c}^{\gamma _1, \gamma _2}\delta )=2\xi f_{m,c}^{\gamma _1, \gamma _2}\delta \), with
where in the last step we used that
Let us consider an aS metric of the form \(g=f_{m,c}^{\gamma _1, \gamma _2}\delta +p\), where \(p=O(r^{-2-\epsilon })\), which satisfies (18) with \(\epsilon \ge 0\).
Proposition 26
The vector field X is almost conformal with respect to g in the sense that
Proof
A direct computation shows that (C3) holds with \(B={\mathcal {L}}_Xp-2\xi p\). Note however that
and the result follows given that \(X=O(r)\). \(\square \)
We now take \(\{e_1,e_2\}\) to be a local orthonormal frame on \(S^2_\rho (a)\) and \(\nu \) its outward unit normal vector. Recall that \(W=\nabla \nu \) is the shape operator of \(S^2_{\rho }(a)\) and \(\Pi =HI-W\) denotes its Newton tensor. If \(X^{\top }=X-\langle X,\nu \rangle \nu \) is the tangential component of X, then
and we obtain from (C3) that
Since
which is easily verified if we take the frame to be principal with respect to the shape operator W, this simplifies to
Thus, summing over A and using that \(H_{a,\rho }^2-\vert W\vert ^2=2K_{a,\rho }\), we obtain
In order to make use of this identity, which first appeared in [62], we need to determine the asymptotics of \(X^\top \).
Proposition 27
One has \(X^\top ={O(\rho ^{-1-\epsilon })}\).
Proof
Recalling that \({\mathfrak {r}}=(x-a)/\rho \), so that \(X=\rho \mathfrak r_i\partial /\partial x_i\), one computes
so that
Thus,
and the result follows. \(\square \)
We now observe that by (30) we may rewrite (54) as
so that
where we used that \({{B=O(\rho ^{-2})}}\). Also, the left-hand side of (C4) may be treated similarly. Indeed, by Proposition 27,
Putting all the pieces of our computation together and using (C6) we get
The proof of Proposition 13 is completed if we note that by (C1) and (C6),
Appendix D: The proof of Proposition 19
Here we indicate how the argument in [26, Appendix F] may be used to prove Proposition 19. In fact, this method allows us to approach the problem in the category of manifolds considered in Definition 9.
Proposition 28
If (M, g) is an asymptotically flat 3-manifold with a non-compact boundary satisfying the RT condition then
Corollary 29
There holds
The key ingredient in the proof is an integral identity derived from the fact that \(S^2_{\rho ,+}(b)\) is a free boundary CMC surface with mean curvature \(2/\rho \) with respect to the metric \(\delta ^+\). In the following, for convenience we shall omit the area element of \(S^2_{\rho ,+}(b)\) and the line element of \(S^{1}_{\rho }(b):=\partial S^2_{\rho ,+}(b)\) in the respective integrals.
Proposition 30
There holds
where \({\mathfrak {r}}=(x-b)/\rho \).
Proof
Apply the identity that follows from equating the right-hand sides of (B2) and (B4) with \(\mu =-\partial /\partial x_3\) to the vector field \(Y_{(\alpha )}=(x_\alpha -b_\alpha ) e^+_{ij}\mathfrak r_i\partial /\partial x_j\) by taking into account that
\(\square \)
We now recall the expansion
where the remainder satisfies \(E={O}(\rho ^{-1-2\tau })\) and \(E^{({-1}')}=O(\rho ^{-2-2\tau })\); see [20, Lemma 2.1]. This reduces to (55) if we take \(e^+=2mr^{-1}\delta ^++{O}(r^{-2})\), which provides the link between Propositions 28 and 19. It follows that
where Proposition 30 has been used to make sure that only those terms which are linear in \({\mathfrak {r}}\) survive in the right-hand side. We now observe that under the decay assumptions (including Regge–Teitelboim) the integrals
and
are \(O(\rho ^{-\tau })\), the same happening to the boundary integrals
Thus, we end up with
Comparing the right-hand side of the above with the definitions of \({\mathfrak {m}}\) and \({\mathcal {C}}^+\), the proof of Proposition 28, and hence of Proposition 19, follows.
Remark 11
The upshot of Corollary 29 is another expression for the center of mass \({\mathcal {C}}^+\), besides (28), derived from Hamiltonian methods, and the isoperimetric one appearing in Theorems 9 and 10. Another rendition of this invariant comes from [18, Theorem 2.4], this time in terms of certain asymptotic flux integrals involving the Einstein tensor of the metric in the interior and the Newton tensor along the boundary; see also [63]. It is remarkable indeed that this kind of invariant admits so many distinct manifestations.
Appendix E: The uniqueness of the free boundary CMC hemispheres
The very last piece of the argument leading to Theorem 10 uses the appropriate uniqueness of the free boundary CMC hemispheres in Theorem 9. Here we justify this step by following the reasoning in [5, Section 4]. We know from the analysis in Sect. 4 that for each \(\rho \) large enough the corresponding hemisphere is a strictly stable free boundary CMC surface graphically described by a function \(\phi _\rho \) on \(S^2_{\rho ,+}({\mathcal {C}}^+)\) satisfying the bound
where \(C>0\) is an absolute constant and the weighted Hölder norm is defined as in the left-hand side of (51). The uniqueness claim is that, for \(\rho \) large enough depending only on C, any other free boundary CMC hemisphere with the same mean curvature and which is graphed by a function satisfying this Hölder bound should coincide with (the graph of) \(\phi _0:=\phi _\rho \). Indeed, assume there exists another such hemisphere, say associated to a function \(\phi _1\). As in [5, Proposition 2.1], the asymptotic roundness of the graphs means that we may interpolate between the corresponding embeddings by setting
for some function \(u(x)=\langle \vec {{\mathfrak {a}}},\nu (x)\rangle +q(x)\), where \(\vec {{\mathfrak {a}}}\in {\mathbb {R}}^2\) is a vector and \(q=O(\rho ^{-1})\). A crucial remark at this point is that all of these surfaces are free boundary (with a possibly non-constant mean curvature \(H_{F_t}\) for \(0<t<1\)) and may be graphed by using functions satisfying the same Hölder bound as \(\phi _0\). Since \(H_{F_0}=H_{F_1}\), the variational vector field \(Y=F_1-F_0\) satisfies
where we used (70) applied to \(dH_{F_0}=\mathscr {L}_{F_0}\), the Jacobi operator associated to \(F_0\), and the fact that \(\Vert d^2H_{F_t}\Vert =O(\rho ^{-3})\) uniformly in t. Thus, there exists an absolute constant \(C_2>0\) such that \(\vert Y\vert \le C_2\) implies \(Y=0\). We next check that \(\vert Y\vert \) (equivalently, \(\vert \vec {{\mathfrak {a}}}\vert \)) may be chosen small enough so as to fulfill this vanishing criterion if \(\rho \) is large. We first note that, again because \(H_{F_0}=H_{F_1}\),
As in [64, Proposition 16] we compute that
where \(Y^\top \) is the tangential component of Y. Starting with (C5) we obtain \(\vert Y^{\top }\vert =O(\rho ^{-3})\) and hence \(\vert Y^{\top }H_{F_t}\vert =O(\rho ^{-4})\). Combining this with (64) we see that the right-hand side of (E1) is \(O(\rho ^{-4})\). On the other hand, since \(\langle \vec {{\mathfrak {a}}},\nu \rangle \) is an approximate eigenfunction of \({\mathscr {L}}_{F_0}\) under Neumann boundary condition with eigenvalue close to \(6m\rho ^{-3}\), the left-hand side of (E1) is \(\ge C_3\vert \vec {{\mathfrak {a}}}\vert \rho ^{-3}\). Thus, \(\vert \vec {{\mathfrak {a}}}\vert \le C_4\rho ^{-1}\) and the uniqueness claim follows provided we take \(\rho \ge C_2^{-1}C_4\).
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Almaraz, S., Lima, L.L.d. Mass, center of mass and isoperimetry in asymptotically flat 3-manifolds. Calc. Var. 62, 196 (2023). https://doi.org/10.1007/s00526-023-02519-1
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DOI: https://doi.org/10.1007/s00526-023-02519-1