Foliation by free boundary constant mean curvature leaves

Foliation by free boundary constant mean curvature leaves

Fabio Montenegro
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 .

For small , we have

and so

The expansion formula (3) now follows immediately by applying the expansion of in [1, p.1604].

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 .

By the Proposition 16 in [3, p.14] we have

and

where is the Jacobi operator.

In particular

(11)

where is the standard Laplace operator in .

Lemma 3.3

We have

(12)

and

(13)

where was defined in (8).

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)

Proof. From Lemma 3.1 and the fact that we have

where

and

Hence (14) follows.

Lemma 3.5

If is the solution of the Neumann problem

(15)

then

(16)

and

(17)

for .

Proof. The function is the solution of the problem (15). Now we can use (12), (13) and the fact

for all interge , to prove (16) and (17).

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