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Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

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.

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References

  1. 1.

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

  2. 2.

    Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. 18(4), 821–830 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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)

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. arXiv:1209.1165

  5. 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)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Kazdan, J., Warner, F.: Prescribing curvatures. In: Differential Geometry. Proc. Sympos. Pure Math., vol. 27, pp. 309–319. Am. Math. Soc., Providence (1975)

    Chapter  Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

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

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Meeks, W., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179(2), 151–168 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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. 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

    Google Scholar 

  14. 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)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Shen, Y., Zhu, S.: Rigidity of stable minimal hypersurfaces. Math. Ann. 309(1), 107–116 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra (1983), vii+272 pp.

    MATH  Google Scholar 

Download references

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.

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Correspondence to Lucas C. Ambrozio.

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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 :xΣf(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

$$\begin{aligned} \partial_{t} = \frac{\partial f}{\partial t} \quad \text{and} \quad \partial_{i} = \frac{\partial f}{\partial x_i} , \end{aligned}$$

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

$$ \partial_{t}=\partial_{t}^T + v_t N_t, $$

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

$$\begin{aligned} \partial_t g_{ij} = & g(\nabla_{\partial_i} \partial_{t},\partial_{j}) + g(\partial_i,\nabla_{\partial_{j}} \partial_t), \\ \partial_t g^{ij} = & -2g^{ik}g^{jl}g(\nabla_{\partial_k} \partial_{t},\partial_l). \end{aligned}$$

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,

$$(\det U)'=\det (U) \text{tr}(U'),$$

we deduce:

Proposition 14

The first variation of area is given by

$$ \frac{d}{dt}|\varSigma_t| = \int_{\varSigma} H_t v_t dA_t + \int_{\partial \varSigma} g\biggl(\nu_t,\frac{\partial f}{\partial t}\biggr)dL_{t}. $$

Proof

Observe that

$$\begin{aligned} \partial_{t}\sqrt{\det[g_{ij}]} = & \frac{1}{2}g^{ij}\partial_{t}g_{ij}\sqrt{\det[g_{ij}]} \\ = & g^{ij}g(\nabla_{\partial_i} \partial_{t},\partial_{j})\sqrt{\det[g_{ij}]} \\ = & (g^{ij}g(\nabla_{\partial_i} \partial_{t}^T,\partial_{j}) + g^{ij}g(\nabla_{\partial_i} N_t,\partial_{j})v_t)\sqrt{\det[g_{ij}]} \\ = & (\text{div}_{\varSigma_t}\partial_t^T + H_{t}v_t)\sqrt{\det[g_{ij}]}. \end{aligned}$$

The first variation formula of area follows. □

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

Proposition 15

$$\begin{aligned} \nabla_{\partial_{i}} N_t = & g^{kl}B_{il}\partial_{k} , \\ \nabla_{\partial_{t}} N_t = & \nabla_{(\partial_{t})^T} N_t - \nabla^{\varSigma_t} v_t, \end{aligned}$$

whereΣ 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

$$ \nabla_{\partial_{t}} N_{t} = g^{ik}g(\nabla_{\partial_{t}} N_{t} , \partial_{k})\partial_{i} = -g^{ik}g(N_t, \nabla_{\partial_{t}} \partial_{k})\partial_{i} = -g^{ik}g(N_t,\nabla_{\partial_{k}}\partial_{t})\partial_{i}. $$

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

$$ g^{ik}g(N_t,\nabla_{\partial_{k}} (v_tN_t))\partial_{i} = (g^{ik}\partial_{k} v_{t})\partial_{i} = \nabla^{\varSigma_t} v_t. $$

Therefore,

$$ \nabla_{\partial_{t}} N_t = \nabla_{(\partial_{t})^T} N_t - \nabla^{\varSigma_t} v_t. $$

 □

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

$$ g(R(U,V)N_t,W) = (\nabla^{\varSigma_t}_{U}B)(V,W) - (\nabla^{\varSigma_t}_{V}B)(U,W). $$

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

$$ \operatorname{Ric}(V,N_t) = g^{ik}(\nabla^{\varSigma_t}_{\partial_{i}}B)(V,\partial_{k}) - dH_t(V), $$

for every V tangent to Σ t .

Proposition 16

The variation of the mean curvature is given by

$$ \partial_t H_t = dH_t (\partial_{t}^{T}) - L_{\varSigma_t}v_t, $$

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

Proof

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

$$\begin{aligned} \partial_t H_t = & \partial_t g^{ij}g(\nabla_{\partial_{i}} N_t, \partial_{j}) + g^{ij}g(\nabla_{\partial_t}\nabla_{\partial_{i}} N_t,\partial_{j}) + g^{ij}g(\nabla_{\partial_i} N_t, \nabla_{\partial_t} \partial_{j}) \\ = & -2g^{ik}g^{jl}g(\nabla_{\partial_{k}}\partial_{t},\partial_{l})g(\nabla_{\partial_{i}} N_t, \partial_{j}) + g^{ij}g(R(\partial_{t},\partial_{i})N_t,\partial_{j}) \\ &{} + g^{ij}g(\nabla_{\partial_i} \nabla_{\partial_t} N_t, \partial_{j}) + g^{ij}g(\nabla_{\partial_{i}} N_t, \nabla_{\partial_{j}}\partial_{t}) \\ = & -2g^{ik}g(\nabla_{\partial_k} \partial_{t},\nabla_{\partial_{i}} N_{t}) - \operatorname{Ric}(\partial_t,N_t) \\ &{} + g^{ij}g(\nabla_{\partial_i} \nabla_{\partial_t} N_t, \partial_{j}) + g^{ij}g(\nabla_{\partial_{i}} N_t, \nabla_{\partial_{j}}\partial_{t}) \\ = & -g^{ij}g(\nabla_{\partial_{i}} N_t, \nabla_{\partial_{j}}\partial_{t}) - \operatorname{Ric}(\partial_{t},N_t) \\ &{} +g^{ij}g(\nabla_{\partial_i} (\nabla_{\partial_t^T} N_t), \partial_{j}) - g^{ij}g(\nabla_{\partial_i}(\nabla^{\varSigma_t} v) ,\partial_{j}). \end{aligned}$$

Now we use the contracted Codazzi equation:

$$\begin{aligned} \operatorname{Ric}(\partial_t ^{T}, N_t) = & g^{ij}(\nabla^{\varSigma_t}_{\partial_{i}}B)(\partial_{t}^T,\partial_{j}) -dH(\partial_t ^T)\\ = & g^{ij}\partial_{i} g(\nabla_{\partial_{t}^T} N_t,\partial_{j}) - g^{ij}g(\nabla_{(\nabla_{\partial_{i}}\partial_t^T)^{T} } N_t,\partial_{j}) \\ &{} -g^{ij}g(\nabla_{\partial_t ^T} N_t,(\nabla_{\partial_{i}} {\partial_{j}})^T) - dH(\partial_t ^T) \\ = & g^{ij}(\partial_{i} g(\nabla_{\partial_{t}^T} N_t,\partial_{j}) - g(\nabla_{\partial_t ^T} N_t,\nabla_{\partial_{i}} {\partial_{j}})) \\ &{} -g^{ij}g(\nabla_{\partial_{j}}N_t,(\nabla_{\partial_{i}}\partial_{t}^T)^T) - dH(\partial_t ^T) \\ = & g^{ij}g(\nabla_{\partial_{i}} (\nabla_{\partial_{t}^T} N_t) , \partial_{j}) - g^{ij}g(\nabla_{\partial_{j}} N_t, \nabla_{\partial_{i}}\partial_{t}^T)-dH(\partial_{t}^T). \end{aligned}$$

Hence, canceling out the corresponding terms, we have

$$\begin{aligned} \partial_t H_t = & - g^{ij}g(\nabla_{\partial_{i}} N_t, \nabla_{\partial_{j}}N_t)v_t - \operatorname{Ric}(N_t,N_t)v_t \\ &{} + dH(\partial_t ^T) - g^{ij}g(\nabla_{\partial_i}(\nabla^{\varSigma_t} v_t) ,\partial_{j}). \end{aligned}$$

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

$$\begin{aligned} (\partial_t H_t)|_{t=0} = -L_{\varSigma_0}v_0 \quad\textit{and} \quad \partial_{t}g(N_t,X)|_{t=0} = -\frac{\partial v_0}{\partial \nu_0} + g(N_0,\nabla_{N_0} X)v_0. \end{aligned}$$

Proposition 18

If each Σ t is a constant mean curvature free boundary surface, then

$$\begin{aligned} \partial_t H_t = - L_{\varSigma_t}v_{t} \quad \textit{and}\quad \frac{\partial v_{t}}{\partial {\nu_{t}}} = g(N_{t},\nabla_{N_{t}}X)v_{t}. \end{aligned}$$

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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

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Keywords

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

Mathematics Subject Classification

  • 53A10
  • 53C24