## Abstract

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru (Splitting of 3-manifolds and rigidity of area-minimizing surfaces, arXiv:1107.5346, 2011). We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

This is a preview of subscription content, access via your institution.

## References

- 1.
Bray, H.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Thesis, Stanford University (1997)

- 2.
Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom.

**18**(4), 821–830 (2010) - 3.
Cai, M., Galloway, G.: Rigidity of area-minimizing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom.

**8**(3), 565–573 (2000) - 4.
Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. arXiv:1209.1165

- 5.
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math.

**33**(2), 199–211 (1980) - 6.
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) - 7.
Kazdan, J., Warner, F.: Prescribing curvatures. In: Differential Geometry. Proc. Sympos. Pure Math., vol. 27, pp. 309–319. Am. Math. Soc., Providence (1975)

- 8.
Ladyzhenskaia, O., Uralt’seva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968), 495 pp.

- 9.
Li, M.: Rigidity of area-minimizing disks in three-manifolds with boundary. Preprint

- 10.
Meeks, W., Yau, S.T.: Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. Math. (2)

**112**(3), 441–484 (1980) - 11.
Meeks, W., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.

**179**(2), 151–168 (1982) - 12.
Micallef, M., Moraru, V.: Splitting of 3-Manifolds and rigidity of area-minimizing surfaces. To appear in Proc. Am. Math. Soc. arXiv:1107.5346

- 13.
Nunes, I.: Rigidity of area-minimizing hyperbolic surfaces in three-manifolds. J. Geom. Anal. (2011) doi:10.1007/s12220-011-9287-8. Published electronically

- 14.
Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature. Ann. Math. (2)

**110**(1), 127–142 (1979) - 15.
Shen, Y., Zhu, S.: Rigidity of stable minimal hypersurfaces. Math. Ann.

**309**(1), 107–116 (1997) - 16.
Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra (1983), vii+272 pp.

## Acknowledgements

I am grateful to my Ph.D advisor at IMPA, Fernando Codá Marques, for his constant advice and encouragement. I also thank Ivaldo Nunes for enlightening discussions about free boundary surfaces. Finally, I am grateful to the hospitality of the Institut Henri Poincaré, where the first drafts of this work were written in October/November 2012. I was supported by CNPq-Brazil and FAPERJ.

## Author information

### Affiliations

### Corresponding author

## Additional information

The author was supported by CNPq-Brazil and FAPERJ.

## Appendix

### Appendix

For completeness we include some general formulae for the infinitesimal variation of some geometric quantities of properly immersed hypersurfaces under variations of the ambient manifold (*M*
^{n+1},*g*) that leave the boundary of the hypersurface inside *∂M*.

We begin by fixing some notation. Let (*M*
^{n+1},*g*) be a Riemannian manifold with boundary *∂M*. Let *X* denote the unit normal vector field along *∂M* that points outside *∂M*.

Let *Σ*
^{n} be a manifold with boundary *∂Σ* and assume *Σ* is immersed in *M* in such way that *∂Σ* is contained in *∂M*. The unit conormal of *∂Σ* that points outside *Σ* will be denoted by *ν*. Given *N* a local unit normal vector field to *Σ*, the second fundamental form is the symmetric tensor *B* on *Σ* given by *B*(*U*,*W*)=*g*(∇_{
U
}
*N*,*W*) for every *U*, *W* tangent to *Σ*. The mean curvature *H* is the trace of *B*. *Σ* is called *minimal* when *H*=0 on *Σ* and *free boundary* when *ν*=*X* on *∂Σ*.

We consider variations of *Σ* given by smooth maps *f*:*Σ*×(−*ϵ*,*ϵ*)→*M* such that, for every *t*∈(−*ϵ*,*ϵ*), the map *f*
_{
t
}:*x*∈*Σ*↦*f*(*x*,*t*)∈*M* is an immersion of *Σ* in *M* such that *f*
_{
t
}(*∂Σ*) is contained in *∂M*.

The subscript *t* will be used to denote quantities associated with *Σ*
_{
t
}=*f*
_{
t
}(*Σ*). For example, *N*
_{
t
} will denote a local unit vector field normal to *Σ*
_{
t
} and *H*
_{
t
} will denote the mean curvature of *Σ*
_{
t
}.

It will be useful for the computations to introduce local coordinates *x*
^{1},…,*x*
^{n} in *Σ*. We will also use the simplified notation

where *i* runs from 1 to *n*. *∂*
_{
t
} is called the variational vector field. We decompose it in its tangent and normal components:

where *v*
_{
t
} is the function on *Σ*
_{
t
} defined by *v*
_{
t
}=*g*(*∂*
_{
t
},*N*
_{
t
}).

First, we look at the variation of the metric tensor *g*
_{
ij
}=*g*(*∂*
_{
i
},*∂*
_{
j
}).

### Proposition 13

### Proof

The first equation is straightforward. The second follows from differentiating *g*
^{ik}
*g*
_{
kl
}=*δ*
_{
il
}. □

From the well-known formula for the derivative of the determinant,

we deduce:

### Proposition 14

*The first variation of area is given by*

### Proof

Observe that

The first variation formula of area follows. □

Next, we look at the variations of the normal field.

### Proposition 15

*where* ∇^{Σ}
*v*
_{
t
}
*is the gradient of the function*
*v*
_{
t
}
*on*
*Σ*
_{
t
}.

### Proof

Since *g*(*N*
_{
t
},*N*
_{
t
})=1, \(\nabla_{\partial_{i}} N_{t}\) and \(\nabla_{\partial_{t}}N_{t}\) are tangent to *Σ*
_{
t
}. The first equation is just the expression of *∂*
_{
i
}
*N*
_{
t
} in the basis {*∂*
_{
k
}}. On the other hand, since *g*(*N*
_{
t
},*∂*
_{
i
})=0, we have

In local coordinates, the gradient of *v*
_{
t
} in *Σ*
_{
t
} is given by \(\nabla^{\varSigma_{t}} v_{t} = (g^{ij}\partial_{j} v_{t})\partial_{i}\). Then we have

Therefore,

□

Before we compute the variation of the mean curvature, let us recall the *Codazzi equation*:

In this equation, *R* denotes the Riemann curvature tensor of (*M*,*g*) and *U*, *V*, and *W* are tangent to *Σ*
_{
t
}.

Taking *U*=*∂*
_{
i
}, *W*=*∂*
_{
k
}, and contracting, we obtain

for every *V* tangent to *Σ*
_{
t
}.

### Proposition 16

*The variation of the mean curvature is given by*

*where*
\(L_{\varSigma_{t}} = \Delta_{\varSigma_{t}} + \operatorname{Ric}(N_{t},N_{t}) + |B_{t}|^{2}\)
*is the Jacobi operator*.

### Proof

Since \(H_{t}= g^{ij}g(\nabla_{\partial_{i}} N_{t}, \partial_{j})\),

Now we use the contracted Codazzi equation:

Hence, canceling out the corresponding terms, we have

The formula follows. □

Finally, we specialize the formulae above in the two particular cases we used in this paper. The proofs are immediate.

### Proposition 17

*If*
*Σ*
_{0}
*is free boundary and* (*∂*
_{
t
})^{T}=0 *at*
*t*=0, *then*

### Proposition 18

*If each*
*Σ*
_{
t
}
*is a constant mean curvature free boundary surface*, *then*

## Rights and permissions

## About this article

### Cite this article

Ambrozio, L.C. Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds.
*J Geom Anal* **25, **1001–1017 (2015). https://doi.org/10.1007/s12220-013-9453-2

Received:

Published:

Issue Date:

### Keywords

- Free boudary minimal surfaces
- Scalar curvature
- Mean curvature
- Rigidity

### Mathematics Subject Classification

- 53A10
- 53C24