Abstract
Let \(\Sigma ^n\) and \(M^{n+1}\) be smooth manifolds with smooth boundary. In this paper, following the techniques developed by White (Indiana Univ Math J 40:161–200, 1991) and Biliotti–Javaloyes–Piccione (Indiana Univ Math J, 1797–1830, 2009), we prove that, given a compact manifold with boundary \(\Sigma ^n\) and a manifold with boundary \(M^{n+1}\), for a generic set of Riemannian metrics on M every free boundary CMC embedding \(\phi :\Sigma \rightarrow M\) is nondegenerate.
Similar content being viewed by others
Notes
More precisely, we assume that \(\Gamma \) is endowed with a Banach manifold structure that makes the inclusion \(\Gamma \hookrightarrow Met^k(M)\) continuous when \(Met^k(M)\) is endowed with the weak Whitney \(C^k\)topology.
Namely, White’s result does not take into consideration the degeneracy of iterated closed geodesics.
For definition and properties of Whitney \(C^k\)topology see [16].
References
Abraham, R.: Bumpy metrics. In: Global analysis. Proc. Sympos. Pure Math., vol. 14, pp. 1–3. Berkeley (1970)
Ambrosio, L.: Lecture notes on elliptic partial differential equations. Unpublished lecture notes, Scuola Normale Superiore di Pisa (2015)
Ambrozio, L.: Rigidity of areaminimizing free boundary surfaces in mean convex threemanifolds. J. Geom. Anal. 25(2), 1001–1017 (2015) (14, 38)
Anosov, D.V.: Generic properties of closed geodesics. Izv. Akad. Nauk SSSR Ser. Mat. 46(4), 675–709, 896 (1982)
Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Zeitschrift 185, 339–353 (1984)
Bettiol, R., Giambò, R.: Genericity of nondegenerate geodesics with general boundary conditions. Topol. Methods Nonlinear Anal 35(2), 339–365 (2010) (22)
Bettiol, R.G., Piccione, P., Santoro, B.: Deformations of free boundary CMC hypersurfaces and applications. J. Geom. Anal. 27(4), 3254–3284 (2017) (23, 69)
Biliotti, L., Javaloyes, M.A., Piccione, P.: Genericity of nondegenerate critical points and morse geodesic functionals. Indiana Univ. Math. J., 1797–1830 (2009)
Biliotti, L., (IPARM), Javaloyes, M.A., (EGRANSGT), Piccione, P., (BRSPL): On the semiRiemannian bumpy metric theorem (English summary). J. Lond. Math. Soc. 84(1), 1–18 (2011)
Biliotti, L., Mercuri, F., Piccione, P.: On a Gromoll–Meyer type theorem in globally hyperbolic stationary spacetimes. Commun. Anal. Geom. 16, 333–393 (2008) (MR2425470)
Giambò, R., Giannoni, F., Piccione, P.: Genericity of nondegeneracy for light rays in stationary spacetimes. Commun. Math. Phys. 287(3), 903–923 (2009)
Gohberg And Krein: The basic propositions on defect numbers and indices of linear operators. Trans. Am. Math. Soc. 13, 185–264 (1960)
Lang, S.: Differential and riemannian manifolds In: Graduate Texts in Mathematics, vol. 160, p. 185 (1995)
Maximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex threemanifolds. J. Differ. Geom. 106(1), 139–186 (2017) (15)
Ros, A., Vergasta, E.: Stability for hypersurfaces of constant mean curvature with free boundary. Geom. Dedic. 56(1), 19–33 (1995)
Serrano, F.: Whitney topology and normality. Topol. Appl. 52(1), 59–67 (1993)
Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)
White, B.: The space of minimal submanifolds for varying Riemannian metrics. Indiana Univ. Math. J. 40, 161–200 (1991)
Funding
This work was partially funded by the Universidad Industrial de Santander.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
In this appendix, we will give some definitions and important results of functional analysis, which were necessary during the development of the whole paper.
Definition 6.1
If \(f:N\rightarrow M\) is a smooth map and \(S\subset M\) is an embedded submanifold, we say that f is transverse to S if, for every \(p\in f^{1}(S)\), \(T_{f(p)}M=T_{f(p)}S+df_p(T_pN)\).
Definition 6.2
The definition of transversality between a map \(F:\mathcal {X}\rightarrow \mathcal {Y}\) and \(\mathcal {Z} \subset \mathcal {Y}\) a smooth submanifold, where \(\mathcal {X}\) and \(\mathcal {Y}\) are Banach manifolds, is that presented in the Definition 6.1 but with the additional assumption that \(dF_{\mathfrak {x}_0}^{1}(T_{F(\mathfrak {x}_0)}\mathcal {Z})\) is a complemented subspace of \(T_{\mathfrak {x}_0}\mathcal {X}\), i.e., there is a subspace \(\mathcal {V}\subset T_{\mathfrak {x}_0}\mathcal {X}\) such that \(T_{\mathfrak {x}_0}\mathcal {X}=dF_{\mathfrak {x}_0}^{1}(T_{F(\mathfrak {x}_0)}\mathcal {Z})\oplus \mathcal {V}\).
Definition 6.3
If A is a bounded operator in a Hilbert space H and \(\{e_i:i\in I\}\) is a orthonormal bases for H, is defined the Hilbert–Schmidt norm as \(A_{HS}^2=Tr(A^*A)=\sum _{i\in I}Ae_i_H^2\), where \(\cdot _H\) is the norm of H.
Definition 6.4
The Hölder space \(C^{j,\alpha }(\Omega )\), where \(\Omega \) is an open subset of some Euclidean space and \(j\ge 0\) an integer, consists of those functions on \(\Omega \) having continuous derivatives up to order j and such that the jth partial derivatives are Hölder continuous with exponent \(\alpha \), where \(0<\alpha \le 1\). A realvalued function f on ndimensional Euclidean space is Hölder continuous, when there are nonnegative real constants c, such that
Definition 6.5
A linear continuous operator \(T:E\rightarrow F\) between normed spaces is Fredholm if \(\mathrm {Ker}\ T\) is finite dimensional and \(\mathrm {Im}\ T\) is close and finite codimensional, the index of T is \(\mathrm {ind}\ T=\mathrm {dim}\ \mathrm {Ker}\ T  \mathrm {dim}\ \mathrm {coker}\ T\). A Fredholm map is a \(C^1\) map \(f: M \rightarrow N\), M and N being differentiable Banach manifolds, such that for each \(x\in M\), the derivative \(df_x: T_x(M) \rightarrow T_{f(x)}(N)\) is a Fredholm operator. The index of f is defined to be the index of \(df_x\) for some x. The definition does not depend on x, see [12].
Theorem 6.1
[13, The Inverse Mapping Theorem, 5.2] Let E, F Banach spaces, U an open subset of E, and Let \(f:U\rightarrow F\) a \(C^p\)morphism with \(p\ge 1\). Assume that for some point \(x_0\in U\) The derivative \(f'(x_0):E \rightarrow F\) is a toplinear isomorphism. Them f is a local \(C^p\)isomorphism at \(x_0\).
Theorem 6.2
[13, The Implicit Mapping Theorem, 5.9] Let U, V be open sets in Banach Spaces E, F respectively, and set \(f:U\times V\rightarrow G\) be a \(C^p\) mapping. Let \((a,b)\in U\times V\), and assume that \(D_2f(a,b):\mathbf{F}\rightarrow G\) is a isomorphism. Let \(f(a,b)=0\). Then there exist a continuous map \(g:U_0\rightarrow V\) defined on an open neighborhood \(U_0\) of a such that \(g(a)=0\) and such that \(f(x,g(x))=0\) for all \(x\in U_0\). If \(U_0\) is taken to be a sufficiently small ball, then g is uniquely determined, and is also of class \(C^p\).
Theorem 6.3
(Local Form of the Submersions) Let X and Y be Banach spaces and let \(f:X\rightarrow Y\) be a submersion in \(x_0\), e.i, \(df(x_0):T_{x_0}\rightarrow T_{f(x_0)}Y\) is surjective and \(\text {Ker}(df(x_0))\) is complemented. Then, there are open sets \(U\subset X\) and \(V\subset \text {Ker}(df(x_0))\), with \(x_0\in U\) and \(0\in V\), and a diffeomorphism \(\varphi : V\times W \rightarrow U\), \(W\subset Y\) closed subspace, such that \(f\circ \varphi (x,w)=w\), for all \((x,w)\in V\times W\)
Appendix B
In this appendix, we prove some lemmas of linear algebra in spaces of infinite dimension that are necessary in the proof of the genericity of the Bumpy Metrics in Sect. 3
Lemma 7.1
Let V be a infinite dimensional vector space with inner product \(\langle , \rangle \) and \(W\subset V\) an finite dimensional subspace. Let us suppose that \(V=W\oplus W^\bot \). Let \(Z\subset V\) be a subspace, \(Z\supset W^\bot \). Then

1.
\((W^\bot )^\bot =W\).

2.
\(V=Z\oplus Z^\bot \)
Proof

1.
Clearly \(W\subset (W^\bot )^\bot \). Now, if \(w\in (W^\bot )^\bot \), we can write \(w=w_1+w_2\), with \(w_1\in W\) and \(w_2\in W^\bot \). From the fact \(W\subset (W^\bot )^\bot \), we have \(w_1\in (W^\bot )^\bot \). Thus \(w_2=w+w_1 \in (W^\bot )^\bot \) (recall that \((W^\bot )^\bot \) is a closed subspace of V). Whence \(w_2\in W^\bot \cap (W^\bot )^\bot =\{0\}\), that is, \(w=w_1\in W\). So, we conclude that \((W^\bot )^\bot \subset W\).

2.
Note that the quotient spaces \(Z/W^\bot \) and \(V/W^\bot \) are both finite dimensional and \(Z/W^\bot \subset V/W^\bot \). Given \(v\in V\), there are unique \(v_1\in W\) and \(v_2\in W^\bot \) such that \(v=v_1+v_2\). Therefore, the map \(\pi _1:V/W^\bot \rightarrow W\), \(\pi _1(v+W^\bot )=v_1\), is an isomorphism. Let \(i:Z/W^\bot \hookrightarrow V/W^\bot \) be the inclusion map. Consider \(L=\pi _1\circ i:Z/W^\bot \rightarrow W\) and let \(\widetilde{Z}=\text {Im}L\subset W\). Thus, \(\widetilde{Z}=\{ zz_2 : z\in Z, z=z_1+z_2, z_1\in W,z_2\in W^\bot \}\subset Z\) and \(W=\widetilde{Z}\oplus \widetilde{Z}^\bot \). Let \(\{x_1,\ldots ,x_k \}\subset W\) be a basis for \( \widetilde{Z}^\bot \). We claim that \(x_1,\ldots ,x_k\in Z^\bot \). Let \(z+W^\bot \in Z/W^\bot \), \(z=z_1+z_2\in Z\), \(z_1\in W\) and \(z_2\in W^\bot \). We have
$$\begin{aligned} \langle z, x_i\rangle&= \langle z, x_i\rangle + \langle W^\bot , x_i\rangle =\langle z+W^\bot , x_i\rangle \\&= \langle z_1+z_2+W^\bot , x_i\rangle = \langle z_1+W^\bot , x_i\rangle \\&= \langle z_1, x_i\rangle + \langle W^\bot , x_i\rangle =0, \end{aligned}$$since \(z_1\in \widetilde{Z}\), \(x_i\in \widetilde{Z}^\bot \subset W\), \(i=1,\ldots ,k\).
Now we prove that \(\text {Span}\{Z,x_1,\ldots ,x_k\}=V\). Let \(v=v_1+v_2\in V\) be given, with \(v_1\in W\) and \(v_2\in W^\bot \subset Z\). There holds that \(v_1= \widetilde{z}_1 + \sum _{i=1}^k a_ix_i\), \(\widetilde{z}_1\in \widetilde{Z}\), where \(a_i\), \(i=1,\ldots ,k\), are scalars. Whence \(v=(\widetilde{z}+v_2)+\sum _{i=1}^k a_ix_i\).
It remains to show that \(\text {Span}\{x_1,\ldots ,x_k\}=Z^\bot \). In fact, it suffices to show that \(\text {Span}\{x_1,\ldots ,x_k\}\supset Z^\bot \). Let
be the inclusion map, and
the natural projection. We have that \(\text {Ker}(\pi _2\circ i)=\{0\}\), so \(\pi _2\circ i\) is injective, thus \(\text {Dim}(Z^\bot )\le k\). \(\square \)
Lemma 7.2
Let V be a infinitedimensional vector space with inner product \(\langle , \rangle \) and \(W\subset V\) an finite dimensional subspace. Suppose that \(V=W\oplus W^\bot \). Let \(Z\subset V\) be a subspace, \(Z\supset W^\bot \), such that for all \(w\in W{\setminus }\{0\}\), there exist \(z\in Z\) with \(\langle z,w\rangle \ne 0\). Then \(Z=V\)
Proof
We show that
First suppose that (\(\forall w\in W{\setminus }\{0\})(\exists z\in Z)(\langle z,w\rangle \ne 0\). Let \(w\in W\cap Z^\bot \) be given, then \(\forall z \in Z\), \(\langle z,w\rangle = 0\), so \(w=0\), that is, \(W\cap Z^\bot =\{0\}\).
Now suppose that \(W\cap Z^\bot =\{0\}\) and let \(w\in W{\setminus } \{0\}\), then \(w\notin Z^\bot \), so there is \(z\in Z\) such that \(\langle z,w\rangle \ne 0\).
Since \(Z\supset W^\bot \), by 1) of Lemma 7.1 we have \(Z^\bot \subset (W^\bot )^\bot = W\). From the fact \(W\cap Z^\bot =\{0\}\), we deduce that \(Z^\bot =\{0\}\). Also by 2) of Lemma 7.1 we have \(V=Z\oplus Z^\bot \). So, \(V=Z\). \(\square \)
Next lemma were taken from [8, Lemma 2.2].
Lemma 7.3
Let \(L : U \rightarrow V\) be a linear map between vector spaces, and let \(S\subset V\) be a subspace of finite codimension. Then \(L^{1}(S)\) is finite codimensional in U, and
.
Rights and permissions
About this article
Cite this article
Cárdenas, C.W.R. Genericity of Nondegenerate Free Boundary CMC Embeddings. Mediterr. J. Math. 17, 188 (2020). https://doi.org/10.1007/s00009020016161
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009020016161