# Foliation by free boundary constant mean curvature leaves

###### Abstract

Let be a Riemannian manifold of dimension with smooth boundary and . We prove that there exists a smooth foliation around whose leaves are submanifolds of dimension , constant mean curvature and its arrive perpendicular to the boundary of M, provided that is a nondegenerate critical point of the mean curvature function of .

## 1 Introduction

The strategy of the proof of this result was inspired by [2]. In this work, Rugang Ye considered the foliation by geodesic spheres around of small radius and showed that this foliation can be perturbed into a foliation whose leaves are spheres of constant mean curvature, provided that is a nondegenerate critical point of the scalar curvature function of . So we are going to consider a family of foliations whose leaves are submanifolds of with boundary contained in and it’s arriving perpendicular to the boundary of . The idea is then to perturb each leaves to obtain, via implicit function theorem, a foliation whose leaves are hemispheres of constant mean curvature and its arrive perpendicular to the boundary of M, provided that is a nondegenerate critical point of the mean curvature function of .

We refer to [1] for basic terminology in local Riemannian geometry. Let be an (n+1)-dimensional Riemannian manifold with smooth boundary , . We will denote by and the covariant derivatives and by and the full Riemannian curvature tensor of and , respectivily. The trace of second fundamental form of the boundary will be denoted by . We will make use of the index notation for tensors, commas denoting covariant differentiation and we will adopt the summation convention.

###### Definition 1.1

Let denote the inward unit normal vector field along . Let be a submanifold with boundary contained in . The unit conormal of that points outside will be denoted by . is called free boundary when on .

Acknowledgements. Part ot this work was done while the author was visiting the university of Princeton, New Jersey. I would like to thank Fernando C. Marques for his kind invitation and helpful discussions on the subject.

## 2 Fermi Coordinate System

Consider a point and an orthonormal basis for . Let be the open ball in . There are and for which we can define the Fermi coordinate system centered at , , given by

(1) |

where is the normal coordinate system in centered at and is the inward unit vector normal to the boundary at .

For each we will consider the Fermi coordinate system centered at

which we denote by , and it defined by

(2) |

where

are the parallel transport of to along the geodesic in , and is the inward unit vector normal to the boundary at .

We will denote the metric tensor of by , the coefficients of in the coordinates system by , and . The expansion of (up to fourth order) in Fermi coordinates can be found in [1, p.1604].

###### Lemma 2.1

In Fermi coordinates centered at we have , , and

(3) |

where every coefficient is computed at and , , , , , and are smooth functions. Here underlined indices mean covariant differentiation as tensor on the boundary.

Proof. We write the expansion

where is the identity matrix and is homogeneous of degree .

## 3 Pertubation by free boundary submanifods

We will work with the following set of functions

(4) |

where .

For we define

and

(5) |

where is the dilation for and sufficiently small such that

There are numbers and such that is an embedded hypersurface in for any and . In addition is a free boundary submanifold of , this is, and its arrive perpendicular to the boundary of M, because . We denote the inward mean curvature function of by .

For we denote the inward mean curvature of the surface at with respect to the metric on , given by , here is the metric tensor in .

For each we have

(6) |

For and we have

But, by the Lemma 2.2 in [1, p.1604],

with when . One readily checks extends smoothly to the euclidean metric when goes to zero. Hence also extends to . Then by a straightforward computation the inward mean curvature function of at with respect to the metric on , can be written as

(7) |

where , ,

(8) |

(9) |

and is the standard Laplace operator on relative to the metric .

###### Lemma 3.1

We have

(10) |

where every coefficient is computed at .

###### Corollary 3.2

The following holds true

Now we consider as a mapping from into and let denote the differential of with respect to . In order to calculate we consider the variation of by smooth maps given by . For each we denote . Note that is an embedded in with . We will denote by a unit vector field normal to and the mean curvature of . We decompose the variational vector field

where is the function on defined by .

In particular

(11) |

where is the standard Laplace operator in .

###### Lemma 3.3

The Jacob operator

has an -dimensional kernel consisting of first order spherical harmonic functions , , which satisfy

In addition we have the -decompositions of spaces and . Let P denote the orthogonal projection from onto , and be the isomorphism sending to , the th coordinate basis. Define , that is,

because

where .

###### Lemma 3.4

We have

(14) |

###### Lemma 3.5

If is the solution of the Neumann problem

(15) |

then

(16) |

and

(17) |

for .

## 4 Main Theorem

###### Definition 4.1

Consider and let be a neighborhood of on . A smooth codimension 1 foliation of for a neighborhood of is called a free boundary foliation centered at , provided that its leaves are all closed and free boundary.

###### Theorem 4.2

If is a nondegenerate critical point of the mean curvature function of , then there exist and smooth functions and with such that for all . Hence the family is a smooth family of constant mean curvature spheres with having mean curvature . Furthermore is a free boundary foliation centered at p.

Proof. We will use the Taylor’s formula with integral remainder

where

We are interested in solving the equation , but first we are going to treat the equation

(18) |

where denotes the orthogonal projection from onto .

By (11) and the fact we can write the equation in (18) as follows (after division by )

where

Consider the mapping given by

where .

For , let be a solution of the equation

One sees that and is a bounded invertible linear transformation. By the implicit function theorem we can solve for a function , for some , with . Furthermore

where

Since

we have, for ,

Then, by Lemma 3.5,

On the other hand,

so that

(19) |

and