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.

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

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

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

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

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

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

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

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