Abstract
We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from (\(\text {n}+1\))-dimensional Euclidean space into a given (\(\text {n}+1\))-dimensional Riemannian manifold which characterizes the existence, stability and dynamics of constant mean curvature submanifolds. This is done in terms of a reduced area function on the Euclidean space, which is given constructively and can be computed perturbatively. This allows us to derive adiabatic and effective dynamics of the bubbles. The results can be mapped by rescaling to the dynamics of fixed size bubbles in almost Euclidean Riemannian manifolds.
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Notes
We choose their orientation to be compatible with that of M.
Like the Feshbach–Schur map, which comes from reconceptulazing of the well-known Feshbach–Schur perturbation theory, the Lyapunov–Schmidt map comes from rethinking the well-known Lyapunov–Schmidt reduction technique.
Let g be the ambient metric with its associated Christoffel symbols, \(\Gamma ^j_{k l}\), and \({\bar{g}}\) be the pull back metric of g onto \(\psi (\mathbb {S}^n)\). In any local coordinate, if \(\psi \) is any immersion from \(\mathbb {S}^n\) to \(M^{n+1}\),
where the Latin indices \(i,j =1,\ldots , n+1\) are for coordinates in the ambient manifold and the Greek ones \(1,\ldots ,n\) are on \(\mathbb {S}^n\) and \(\nu ^j\) is the unit normal vector on \(\psi (\mathbb {S}^n)\). At \(\psi (\omega )\), we see that all geometric quantities are smooth functions of the ambient metric and at most 2 derivatives of the immersion \(\psi \), all evaluated at \(\omega \). We remark that, however, dependence on top (2nd) order derivative is linear and comes from the \(\nabla _\alpha \nabla _\beta \psi \) term above.
References
Alikakos, N., Freire, A.: The normalized mean curvature flow for a small bubble in a Riemannian manifold. J. Differ. Geom. 64(2), 247–303 (2003)
Andrews, B.: Volume-preserving anisotropic mean curvature flow. Indiana Univ. Math. J. 50(2), 783–827 (2001)
Antonopoulou, D.C., Karali, G.D., Sigal, I.M.: Stability of spheres under volume preserving mean curvature flow. Dyn. PDE 7, 327–344 (2010)
Athanassenas, M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72(1), 52–66 (1997)
Athanassenas, M.: Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow. Calc. Var. Partial Differ. Equ. 17(1), 1–16 (2003)
Athanassenas, M., Kandanaarachchi, S.: Convergence of axially symmetric volume-preserving mean curvature flow. Pac. J. Math. 259(1), 41–54 (2012)
Bach, V., Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998)
Bach, V., Chen, T., Fröhlich, J., Sigal, I.M.: Smooth Feshbach map and operator-theoretic renormalization group methods. J. Funct. Anal. 203, 44–92 (2003)
Barbosa, J.L., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197(1), 123–138 (1988)
Bourgain, J.: On a homogenization problem. Preprint (2017)
Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg–Landau equation. SIAM J. Math. Anal. 28(4), 769–807 (1997)
Chen, X., Hilhorst, D., Logak, E.: Mass conserved Allen–Cahn equation and volume preserving mean curvature flow (2009). arXiv:0902.3625v1
Colding, T.H., Minicozzi II, W.P.: Minimal submanifolds. Bull. Lond. Math. Soc. 38, 353 (2006)
Conlon, J.G., Spencer, T.: A strong central limit theorem for a class of random surfaces. Commun. Math. Phys. 325(1), 1–15 (2014)
Conlon, J.G., Spencer, T.: Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. Am. Math. Soc. 366(3), 1257–1288 (2014)
Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc. 126(9), 2789–2796 (1998)
Gage, M.: On an area-preserving evolution equation for plane curves. In: De Turck, D.M. (ed.), Nonlinear Problems in Geometry. Contemporary Mathematics, vol. 51. AMS, Providence, pp. 51–62 (1986)
Gray, A.: The volume of a small geodesic ball of a Riemannian manifold. Mich. Math. J. 20, 329–344 (1973)
Griesemer, M., Hasler, D.: On the smooth Feshbach–Schur map. J. Funct. Anal. 254(9), 2329–2335 (2008)
Hartley, D.: Motion by volume preserving mean curvature flow near cylinders. Commun. Anal. Geom. 21(5), 873–889 (2013)
Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)
Huisken, G., Yau, S.T.: Definition of center of mass for isolated physical systems and unique foliation by stable spheres with constant mean curvature flow. Invent. Math. 124, 281–311 (1996)
Matsen, M.W.: The standard Gaussian model for block copolymer melts. J. Phys. Condens. Matter 14, R21–R47 (2002)
Pacard, F., Xu, X.: Constant mean curvature spheres in Riemannian manifolds. Manuscr. Math. 128(3), 275–295 (2009)
Pizzo, A.: Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian. arXiv:1511.07022
Pizzo, A.: Bose particles in a box II. A convergent expansion of the ground state of the Bogoliubov Hamiltonian in the mean field limiting regime. arXiv:1511.07025
Pizzo, A.: Bose particles in a box III. A convergent expansion of the ground state of the Hamiltonian in the mean field limiting regime. arXiv:1511.07026
Spohn, H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71(5/6), 1081 (1993)
Ye, R.: Foliation by constant mean curvature spheres. Pac. J. Math. 147(2), 381–396 (1991)
Willmore, T.J.: Riemannian Geometry. Oxford University Press, Oxford (1993)
Acknowledgements
Ilias Chenn and I. M. Sigal research is supported in part by NSERC Grant No. NA7901. Ilias Chenn is also partly supported by NSERC CGS D Scholarship. G. Fournodavlos is supported by the EPSRC Grant EP/K00865X/1 on “Singularities of Geometric Partial Differential Equations”. We are grateful to the anonymous referee for useful remarks. I. M. Sigal is grateful to Stephen Gustafson for many illuminating discussions and insights. The author’s discussions over a period of many years with Jürg Fröhlich, Tom Spencer and Herbert Spohn played a crucial role in forming his understanding of adiabatic and effective dynamics and of mathematical physics in general.
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To Jürg, Tom and Herbert with friendship and admiration.
Appendices
Appendix A: Expansions of the Induced Metric and Mean Curvature
We Taylor expand the induced metric and mean curvature of \(\theta _{\rho ,z}\) and estimate the non-linear terms in \(\phi \) in \(H^k\), \(k>\frac{n}{2}+2\). One may consult [18] for a thorough study on such classical expansions, we mostly follow the notation in [24]. We abuse slightly notation by denoting the pushforward of \(\omega \) through \(d\theta _{\rho ,z}\) by the same symbol \(\omega \). For instance, given \(x^i\),\(\partial _i\) a set of coordinates and the associated vector fields, we write the corresponding basis of tangent vector fields on \(S^n_z\): \(\zeta _i:=\lambda (1+\phi )\partial _i\omega +\lambda \partial _i\phi \, \omega \in TS^n_z\).
Lemma A.1
(see [24], Lemmas 2.1 and 2.4) The following expansions are valid:
where R is the Riemann curvature tensor and the remainders \(\mathrm R_{\lambda , z}^{(j)}( \phi ), j=0, 2,\) are local terms depending on \(\phi \) and \(\partial \phi \) and satisfying the estimates
provided \(\Vert \phi \Vert _{H^k} \lesssim 1\) and \(i+|\alpha |\le 2\), and
where \(\text {Ric}_z: \omega \rightarrow \text {Ric}_z(\omega ,\omega )\) (Ric\((\cdot ,\cdot )\) is the Ricci curvature tensor of M), \(M'_{\lambda , z}\phi \), \(N'_{\lambda , z}(\phi )\) and \(H_{\lambda , z}\) are linear non-linear terms in \(\phi \) and in its derivatives up to order two, respectively, and independent of \(\phi \) term, satisfying the estimates:
and similarly for \(N'_{\lambda , z}(\phi ')-N'_{\lambda , z}(\phi )\), for \(i + |\alpha |\le 2\), \(k>\frac{n}{2}+2\). (Above, if \(\phi \) is independent of \(\lambda \) and z, then only the term with \(j= |\beta |=0\) survives on the r.h.s..)
Proof
Both expressions (A.1) and (A.3) can be read from [24]. Bounds (A.4) are immediate by definition. In order to derive estimate (A.6) for the non-linearity \(N_{\lambda , z}(\phi )\) we have to first examine its structure. Recall that the mean curvature of \(\theta _{\rho ,z}\) is given byFootnote 4
where \(g^{ij}\) is the inverse matrix to the metric \(g_{ij}:=g(\zeta _i,\zeta _j)\) and \(\nu \) the outward unit normal vector field on \(\theta _{\rho ,z}\):
Note that \(\nu \) is well defined for \(\lambda \) and \(\Vert \phi \Vert _{H^k}\) appropriately small. Hence, we observe that the non-linear terms in \(\phi \) arising in the expansion of \(H(\theta _{\rho ,z})\) are of the form \(b(\lambda \phi (\omega ),\lambda \partial \phi (\omega ))\lambda \partial ^2\phi (\omega )\), where b(s, t) is a simple function, uniformly bounded together with its derivatives, provided \(|s|\ll 1\) and \(|t|\ll 1\). Using these estimates it is not hard (but somewhat tedious) to show that \(\Vert b(\lambda \phi ,\lambda \partial \phi )\lambda \partial ^2\phi \Vert _{H^{k-2}}\lesssim \lambda ^r\Vert \phi \Vert ^r_{H^k}\) for some \(r\ge 2\), provided \(\Vert \phi \Vert _{H^k}\ll 1\) (here we use the condition \(k>\frac{n}{2}+2\)). This completes the proof of the lemma. \(\square \)
Eq (A.1) implies
where Ric is the Ricci curvature tensor of M and \(\mathrm S_{\lambda , z}^{(0)}, r=0, 2,\) are local terms depending on \(\phi \) and \(\partial \phi \) satisfying the estimates
provided \(\Vert \phi \Vert _{H^k} \lesssim 1\) and \(i+|\alpha |\le 2\). Furthermore, multiplying (A.10) by \(\lambda ^{n}(1+\phi )^{n}\), integrating over \(\mathbb {S}^n\) and taking into account that \(\int _{\mathbb {S}^n}\phi =0\), and using the co-area formula and polar coordinates, we obtain:
Corollary A.2
We have the following expansions
where \(a_n\) denotes the area of the Euclidean unit sphere \(\mathbb {S}^n\) and R is the scalar curvature of M, and
The remainders \(\mathrm Q_{\lambda , z}^{(r)}\) and \(\mathrm T_{\lambda , z}^{(r)}\) above are local expression in \(\phi ,\partial \phi \) satisfying the estimates of the type of (4.2).
Appendix B: Existence of Barycenter
In this section, we show that barycenter exists. That is, we show that Eq. (3.4) has a solution. To begin, we state some useful estimates.
Lemma B.1
For the terms defined in equations (3.2)–(3.3), we have the estimate
Proof
Recall that \(\psi (\omega ) = \exp _z(\rho (\xi )\omega )\), where \(\rho (\xi ) := \lambda (1+\phi _{\lambda ,z} +\xi )\). Using the definitions (3.2), we compute the difference, \(\sigma (\xi )-\sigma \). To this end, it suffices to compute the differences of each factor in (3.2):
where T is the parallel transport from \(\theta _{\lambda ,z}(\omega )\) to \(\psi (\omega )\). Continuing the estimate, we have
Recalling that \(\theta _{\lambda ,z} = \exp _z(\rho (0)\omega )\) and letting \(\omega ^0 := 1\), we find furthermore
Similarly, we write \(f_i(\xi ) - f_i\). Hence, using this, we only need to estimate the following items to complete the proof:
where \(g_p\) is the value of the metric at p, T(s, z) is parallel transport from z along \(\omega \) for time s. Then the difference of quantities in the statement of the question are composition and smooth functions of the above with at most two derivatives of \(\phi _{\lambda ,z}\) and \(\xi \). Note that the first four quantities only depends on the ambient geometry of \(M^{n+1}\). Since we are working lcoally, we may assume that we are working on a compact subset of M. Thus, the first four are smooth functions, the first four quantity exhibits Lipschitz estimates in the difference in their argument. For example,
uniformly on \(M^{n+1}\). The next 3 expressions have the obvious estimate
Finally, the last one follows from the fact \(\nu (\psi )\) is a none singular rational function of \(\psi \). \(\square \)
Now, let
where \(\xi \) is defined by Eq. (3.1).
Lemma B.2
Any \(\psi \) sufficiently close to a CMC \(\theta _{\lambda _0,z_0} = \exp _{z_0}(\lambda _0(1+\phi _{\lambda _0,z_0}))\) can be written in the form (2.2), with (3.1) and \(\lambda \) and z satisfying the equation
Proof
We note that \(P(\lambda _0,z_0, \theta _{\lambda _0,z_0}) = 0\). By the implicit function theorem, it suffices to show that
- (i):
-
P is \(C^1\), and
- (ii):
-
\((\partial _{\lambda ,z} P)(\lambda _0,z_0,\theta _{\lambda _0,z_0})\) is an invertible matrix.
(i) We check, by definition of \(f_i\) and \(\xi \), that they are real functions of \(\lambda \), z and \(\phi _{\lambda ,z}\) up to 1 derivative. Since \(\phi _{\lambda ,z}\) is \(C^1\) in \(\lambda \) and z (cf. Proposition 2.4), we see that \(f_i\) and \(\sigma \) are \(C^1\) in \(\lambda \) and z. Since P is linear in \(\xi \), using the estimate of Proposition 2.4 again, we see that P is \(C^1\) in \(\psi \) as well.
(ii) We compute
since \(\xi = 0\) for \(\psi = \theta _{\lambda _0,z_0}\). To compute \(\partial _\lambda \xi \), we use the fact that, by definition of \(\xi \),
for \(\lambda \) sufficiently close to \(\lambda _0\). (Note that we suppressed the identification \(I_z\) between \(T_zM\) and \(\mathbb {R}^{n+1}\) here as z is not varied.) Taking \(\partial _\lambda \) and evaluate at \(\lambda = 0\), we get the result of
Contracting with \(\nu (\theta _{\lambda _0,\theta _0})\), we get
where the last line follow from Proposition 2.4. To compute \(\partial _z \xi \mid _{\lambda _0,z_0}\), we consider curves \(z(t) : (-\epsilon , \epsilon ) \rightarrow M\) with \(z(0) = z_0\) and \(\dot{z}(0) = v\) for any \(v \in T_{z_0}M\) fixed. Then any variation of
is tangential. Taking derivative with respect to t and setting \(t=0\), we see that the expression (B.11) becomes
Using the fact that varying z is only tangential, it follows that
for any \(v \in T_zM\). By definition of \(f_i\),
It follows that
is invertible by Lemma B.1. \(\square \)
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Chenn, I., Fournodavlos, G. & Sigal, I.M. The Effective Dynamics of the Volume Preserving Mean Curvature Flow. J Stat Phys 172, 458–476 (2018). https://doi.org/10.1007/s10955-018-2041-x
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DOI: https://doi.org/10.1007/s10955-018-2041-x