Compactness analysis for free boundary minimal hypersurfaces

Abstract

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.

Appendices

Appendix A: The second variation for smooth hypersurfaces

Our goal is to derive the second variation of the area functional for arbitrary hypersurfaces with boundary in ${\mathcal {N}}^{n+1}$. For convenience, we assume ${\mathcal {N}}$ is isometrically embedded in some Euclidean space of possibly large dimension ${\mathbb {R}}^d$. Let $\Sigma ^n \hookrightarrow {\mathcal {N}}^{n+1}$ be a smooth properly embedded hypersurface with boundary. We consider a two-parameter family of ambient variations $\Sigma (t,s)$ of $\Sigma =\Sigma (0,0)$ defined by

$$\begin{aligned} \Sigma (t,s)=F_{t,s}(\Sigma ) = \Psi _s \circ \Phi _t(\Sigma ) \end{aligned}$$

where $\Phi _t$ and $\Psi _s$ are the flows generated by compactly supported vector fields X and Z in ${\mathbb {R}}^d$, respectively. We assume both X and Z define vector fields in ${\mathfrak {X}}_\partial$ when restricted to ${\mathcal {N}}$, therefore each $\Sigma (t,s)$ is a properly embedded hypersurface in ${\mathcal {N}}^{n+1}$.

We have

$$\begin{aligned} \left. {\frac{\partial F_{t,s}}{\partial s}}(x)\right| _{t=s=0} = Z(x), \, \left. {\frac{\partial F_{t,s}}{\partial t}}(x)\right| _{t=s=0} = X(x), \, \text {and} \, \left. {\frac{\partial }{\partial t}}{\frac{\partial }{\partial s}}F_{t,s}(x)\right| _{t=s=0} = D_X Z (x). \end{aligned}$$

Computing the second variation of volume as in [33], Chapter 2, §9, we obtain

$$\begin{aligned}&{\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0}\\&\quad = \int _{\Sigma }\sum _{i=1}^{n}\langle D_{\tau _i}D_{X}Z,\tau _i\rangle +\left( \sum _{i=1}^{n}\left\langle D_{\tau _i}X,\tau _i \right\rangle \right) \left( \sum _{j=1}^{n}\left\langle D_{\tau _j}Z,\tau _j \right\rangle \right) \\&\qquad + \sum _{i=1}^n \left\langle \left( D_{\tau _{i}} X\right) ^{\bot _{\Sigma }}, \left( D_{\tau _{i}} Z\right) ^{\bot _{\Sigma }}\right\rangle - \sum _{i,j=1}^n \left\langle \tau _{i}, D_{\tau _{j}} X\right\rangle \left\langle \tau _{j}, D_{\tau _{i}} Z\right\rangle \ d\mathscr {H}^n. \end{aligned}$$

In the above formula, the symbol $\perp _{\Sigma }$ denotes the component that is normal to $\Sigma$ in ${\mathbb {R}}^d$, D denotes the Euclidean Levi-Civita connection, and $\{\tau _i\}_{i=1}^{n}$ is a choice of an orthonormal basis of $T_p\Sigma$ at each point p in $\Sigma$ (the above sums are easily shown to be independent of such choice).

Since we assumed that X and Z belong to ${\mathfrak {X}}_\partial$ when restricted to ${\mathcal {N}}$, decomposing the vectors fields X, Z and $D_{X}Z$ into their components that are tangent and perpendicular to ${\mathcal {N}}$ it is possible to use the Gauss equation for the embedding of ${\mathcal {N}}$ in ${\mathbb {R}}^d$ to rewrite the above formula solely in terms of the geometry of the embedding $\Sigma \hookrightarrow {\mathcal {N}}$:

$$\begin{aligned} {\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0}= & {} \int _{\Sigma } div_{\Sigma } (\nabla _{X}Z) + div_{\Sigma } (X) div_{\Sigma }(Z) \\&+ \sum _{i=1}^{n} \left\langle (D_{e_i}X)^{\bot },(D_{e_i}Z)^{\bot }\right\rangle - \sum _{i=1}^n R_{{\mathcal {N}}}(X,\tau _i,Z,\tau _i)\\&-\sum _{i,j=1}^n \langle \tau _{i}, \nabla _{\tau _{j}} X\rangle \langle \tau _{j}, \nabla _{\tau _{i}} Z)\rangle \ d\mathscr {H}^n, \end{aligned}$$

where $\nabla$ and $R_{\mathcal {N}}$ are the Levi-Civita connection and Riemann tensor of ${\mathcal {N}}$ respectively, $\bot$ denotes the component that is normal to $\Sigma$ and tangent to ${\mathcal {N}}$, and, given any vector field Y that is tangent to ${\mathcal {N}}$, we write $div_\Sigma (Y) = \sum _{i=1}^{n} \langle \nabla _{\tau _i}Y,\tau _i \rangle$.

In the sequel of this Appendix, the Levi-Civita connection of $\Sigma$ will be denoted by ${\nabla }^{\Sigma }$ and ${\nabla }^\bot$ will denote the connection of the normal bundle of $\Sigma$ in ${\mathcal {N}}$. The projections $^\bot$, $^\top$ are the normal and tangential projections for $\Sigma (t,s) \hookrightarrow {\mathcal {N}}$ respectively so that

$$\begin{aligned} X=X^\top + X^\bot \, \text {and} \, Z=Z^\top + Z^\bot \end{aligned}$$

since they are both tangent to ${\mathcal {N}}$. According to that decomposition, we write

$$\begin{aligned} {\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0}= & {} \int _{\Sigma } div_\Sigma ({\nabla }_{X}Z) + {\mathcal {A}}(X^{\bot },Z^{\bot }) \nonumber \\&+ {\mathcal {B}}(X^{\top },Z^{\top }) + {\mathcal {C}}(X^{\top },Z^{\bot }) + {\mathcal {C}}(Z^{\top },X^{\bot }) \ d\mathscr {H}^{n}, \end{aligned}$$

(A.1)

where, using the identity $div_{\Sigma }(Y) = div_{\Sigma } (Y^{\top }) - \langle Y^{\top }, H\rangle$ for H the mean curvature vector of $\Sigma \hookrightarrow {\mathcal {N}}$ and any vector field Y tangent to ${\mathcal {N}}$, the terms ${\mathcal {A}}, {\mathcal {B}}$ and ${\mathcal {C}}$ are given by:

$$\begin{aligned} {\mathcal {A}}(X^{\bot },Z^{\bot }) =&\ \langle X^{\bot }, H\rangle \langle Z^{\bot },H \rangle + \sum _{i=1}^n\left\langle \nabla ^{\bot }_{\tau _i} (X^\bot ) , \nabla ^{\bot }_{\tau _i} (Z^\bot )\right\rangle \nonumber \\&- \sum _{i=1}^{n} R_{\mathcal {N}} (X^\bot ,\tau _i, Z^\bot ,\tau _i) - \sum _{i,j=1}^n \left\langle \tau _i,\nabla _{\tau _j} (X^\bot ) \right\rangle \left\langle \tau _j, \nabla _{\tau _i} (Z^\bot ) \right\rangle , \nonumber \\ {\mathcal {B}}(X^{\top },Z^{\top }) =&\ div_{\Sigma }(X^{\top })div_{\Sigma }(Z^{\top }) + \sum _{i=1}^{n} \left\langle A(X^{\top },\tau _i),A(Z^{\top },\tau _i))\right\rangle \nonumber \\&- \sum _{i=1}^{n} R_{\mathcal {N}} (X^\top ,\tau _i, Z^\top ,\tau _i) - \sum _{i,j=1}^{n} \left\langle \tau _i,\nabla _{\tau _j} (X^\top ) \right\rangle \left\langle \tau _j, \nabla _{\tau _i} (Z^\top ) \right\rangle , \nonumber \\ {\mathcal {C}}(X^{\top },Z^{\bot }) =&- \langle Z^{\bot },H \rangle div_{\Sigma }(X^\top ) + \sum _{i=1}^n \left\langle A(X^{\top },\tau _i), \nabla ^{\bot }_{\tau _i}(Z^{\bot }) \right\rangle \nonumber \\&- \sum _{i=1}^n R_{{\mathcal {N}}}(X^{\top },\tau _i,Z^{\bot },\tau _i) + \sum _{i=1}^n \left\langle A\left( \nabla ^{\Sigma }_{\tau _i} (X^{\top }), \tau _i\right) ,Z^{\bot } \right\rangle . \end{aligned}$$

(A.2)

Using the Gauss equation for $\Sigma \hookrightarrow {\mathcal {N}}$ (valid for all U, V, W, Y tangent to $\Sigma$), namely

$$\begin{aligned} R_{\Sigma }(U,V,W,Y) = R_{{\mathcal {N}}}(U,V,W,Y) + \langle A(U,W),A(V,Y) \rangle - \langle A(U,Y),A(V,W) \rangle , \end{aligned}$$

we can write

$$\begin{aligned} {\mathcal {B}}(X^{\top },Z^{\top }) =&\ div_{\Sigma }(X^{\top })div_{\Sigma }(Z^{\top }) + \langle d (X^{\top })^{\flat } , d (Z^{\top })^{\flat } \rangle - \langle \nabla ^{\Sigma } (X^{\top }), \nabla ^{\Sigma } (Z^{\top }) \rangle \nonumber \\&- Ric_{\Sigma }(X^{\top },Z^{\top }) + \langle A(X^{\top },Z^{\top }),H \rangle . \end{aligned}$$

(A.3)

In the above formula, we use the standard musical notation relating vector fields to their dual one-forms, and d denotes the exterior differential. Defining the one-form $\omega$ in $\Sigma$ by

$$\begin{aligned} \omega (\xi ) = \langle A(Z^{\top },\xi ),X^{\bot }\rangle + \langle A(X^{\top },\xi ),Z^{\bot } \rangle - \langle Z^{\bot }, H \rangle \langle X^{\top }, \xi \rangle - \langle X^{\bot }, H \rangle \langle Z^{\top }, \xi \rangle , \end{aligned}$$

(A.4)

for all vectors $\xi$ tangent to $\Sigma$, a straightforward computation using the Codazzi equation (valid for all for U, V, W tangent to $\Sigma$ and $\eta$ normal to $\Sigma$), namely

$$\begin{aligned} \langle (\nabla _{U}A)(V,W),\eta \rangle - \langle (\nabla _{V}A)(U,W),\eta \rangle = R_{{\mathcal {N}}}(U,V,\eta ,W), \end{aligned}$$

gives

$$\begin{aligned} {\mathcal {C}}(X^{\top },Z^{\bot }) + {\mathcal {C}}(Z^{\top },X^{\bot }) = div_{\Sigma } \omega + \left\langle \nabla ^{\bot }_{X^{\top }} (Z^{\bot }), H \right\rangle + \left\langle \nabla ^{\bot }_{Z^{\top }} (X^{\bot }), H \right\rangle . \end{aligned}$$

(A.5)

Finally,

$$\begin{aligned} div_{\Sigma }(\nabla _{X}Z) =&\ div_{\Sigma }\left( (\nabla _{X}Z)^{\top }\right) - \left\langle \nabla _{X^{\bot }}(Z^{\bot }), H \right\rangle - \langle A(X^{\top },Z^{\top }), H \rangle \nonumber \\&- \left\langle \nabla _{X^{\bot }}(Z^{\top }), H \right\rangle - \left\langle \nabla ^{\bot }_{X^{\top }}(Z^{\bot }), H \right\rangle . \end{aligned}$$

(A.6)

Substituting Eqs. (A.2),(A.3), (A.5) and (A.6) into (A.1), we obtain

$$\begin{aligned}&{\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0} \\&\quad = \int _{\Sigma } \langle \nabla ^{\bot }(X^\bot ) , \nabla ^{\bot }(Z^\bot )\rangle - Ric_{\mathcal {N}} (X^\bot , Z^\bot ) - \langle (X^\bot ),(Z^\bot ) \rangle |A|^2 \\&\qquad + \langle X^{\bot }, H\rangle \langle Z^{\bot },H \rangle - \langle \nabla _{X^{\bot }} (Z^{\bot }), H \rangle - \langle [X^{\bot },Z^{\top }],H \rangle \\&\qquad + div_{\Sigma }(X^{\top }) div_{\Sigma }(Z^{\top }) + \langle d (X^{\top })^{\flat } , d (Z^{\top })^{\flat } \rangle - \langle \nabla ^{\Sigma } X^{\top }, \nabla ^{\Sigma } Z^{\top }\rangle \\&\qquad - Ric_{\Sigma }(X^{\top },Z^{\top }) + div_\Sigma ((\nabla _{X}Z)^{\top }) + div_{\Sigma }\omega \ d\mathscr {H}^{n}. \end{aligned}$$

The final manipulation we perform is to integrate by parts. Choosing a local orthonormal frame $\{\tau _i\}$ geodesic at a point p in $\Sigma$, the following computation at p (where we sum over repeated indices i and j) yields:

$$\begin{aligned} div_{\Sigma }(\nabla ^{\Sigma }_{X^{\top }}(Z^{\top }))= & {} \tau _i \left\langle \nabla ^{\Sigma }_{X^{\top }}(Z^{\top }),\tau _i \right\rangle \\= & {} \tau _i (\left\langle \nabla ^{\Sigma }_{\tau _j}(Z^{\top }),\tau _i \right\rangle \left\langle X^{\top },\tau _j \right\rangle ) \\= & {} \left\langle \nabla ^{\Sigma }_{\tau _i}\nabla ^{\Sigma }_{\tau _j}(Z^{\top }),\tau _i\right\rangle \left\langle X^{\top },\tau _j \right\rangle + \left\langle \nabla ^{\Sigma }_{\tau _j}(Z^{\top }),\tau _i \right\rangle \left\langle \nabla ^{\Sigma }_{\tau _i}(X^{\top }),\tau _j \right\rangle \\= & {} R_{\Sigma }(\tau _i,\tau _j,\tau _i,Z^{\top })\left\langle X^{\top },\tau _j\right\rangle + \left\langle \nabla ^{\Sigma }_{\tau _j} \nabla ^{\Sigma }_{\tau _i}(Z^{\top }),\tau _i\right\rangle \left\langle X^{\top },\tau _j\right\rangle \\&+ \left\langle \nabla _{\tau _j}Z^{\top },\tau _i \right\rangle \left\langle \nabla _{\tau _i}X^{\top },\tau _j \right\rangle \\= & {} Ric_{\Sigma }(X^{\top },Z^{\top }) + d(div_{\Sigma }(Z^{\top }))(X^{\top }) \\&- \left( \left\langle d (X^{\top })^{\flat } , d (Z^{\top })^{\flat } \right\rangle - \langle \nabla ^{\Sigma } X^{\top }, \nabla ^{\Sigma } Z^{\top }\rangle \right) . \end{aligned}$$

Therefore

$$\begin{aligned}&\int _{\Sigma } div_{\Sigma }(X^{\top })div_{\Sigma }(Z^{\top }) + \langle d (X^{\top })^{\flat } , d (Z^{\top })^{\flat } \rangle \\&\qquad - \langle \nabla ^{\Sigma } X^{\top }, \nabla ^{\Sigma } Z^{\top }\rangle - Ric_{\Sigma }(X^{\top },Z^{\top }) \ d\mathscr {H}^{n} \\&\quad = \int _{\Sigma } div_{\Sigma }\left( div_{\Sigma }(Z^{\top })(X^{\top })\right) - div_{\Sigma }\left( \nabla ^{\Sigma }_{X^{\top }}(Z^{\top })\right) \ d\mathscr {H}^{n} \\&\quad = \int _{\partial \Sigma } div_{\Sigma }(Z^{\top })\langle X^{\top }, \nu \rangle - \left\langle \nabla _{X^{\top }}(Z^{\top }),\nu \right\rangle \ d\mathscr {H}^{n-1}, \end{aligned}$$

where $\nu$ denotes the unit outward conormal of $\Sigma$. Thus,

$$\begin{aligned}&{\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0} \nonumber \\&\quad =\int _{\Sigma } \langle \nabla ^{\bot }(X^\bot ) , \nabla ^{\bot }(Z^\bot ) \rangle - Ric_{\mathcal {N}} (X^\bot , Z^\bot ) - |A|^2\langle X^\bot , Z^\bot \rangle \ d\mathscr {H}^n \nonumber \\&\qquad + \int _{\Sigma } \langle X^{\bot }, H\rangle \langle Z^{\bot },H \rangle - \langle \nabla _{X^{\bot }} (Z^{\bot }), H \rangle - \langle [X^{\bot },Z^{\top }],H \rangle \ d\mathscr {H}^n \nonumber \\&\qquad + \int _{\partial \Sigma } \langle {\nabla }_X Z, \nu \rangle + \omega (\nu ) + div_{\Sigma }(Z^{\top })\langle X^{\top }, \nu \rangle - \left\langle \nabla _{X^{\top }}(Z^{\top }),\nu \right\rangle \ d\mathscr {H}^{n-1}. \end{aligned}$$

(A.7)

where $\omega$ is is the one-form defined in (A.4). Since

$$\begin{aligned} \langle \nabla _{X}Z , \nu \rangle + \omega (\nu )= & {} \left\langle \nabla _{X^{\bot }}(Z^{\bot }),\nu \right\rangle + \left\langle \nabla _{X^{\bot }}(Z^{\top }),\nu \right\rangle + \left\langle \nabla _{X^{\top }}(Z^{\bot }),\nu \right\rangle + \left\langle \nabla _{X^{\top }}(Z^{\top }),\nu \right\rangle \nonumber \\&+ \langle A(Z^{\top },\nu ),X^{\bot }\rangle + \langle A(X^{\top },\nu ),Z^{\bot }\rangle - \langle Z^{\bot }, H \rangle \langle X^{\top }, \nu \rangle \nonumber \\&- \langle X^{\bot }, H \rangle \langle Z^{\top }, \nu \rangle \nonumber \\= & {} \left\langle \nabla _{X^{\bot }}(Z^{\bot }),\nu \right\rangle + \langle [X^{\bot },Z^{\top }],\nu \rangle + \left\langle \nabla _{X^{\top }}(Z^{\top }),\nu \right\rangle \nonumber \\&- \langle Z^{\bot }, H \rangle \langle X^{\top }, \nu \rangle - \langle X^{\bot }, H \rangle \langle Z^{\top }, \nu \rangle , \end{aligned}$$

(A.8)

combining (A.7) and (A.8) we obtain:

$$\begin{aligned}&{\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0} \\&\quad =\int _{\Sigma } \langle \nabla ^{\bot }(X^\bot ) , \nabla ^{\bot }(Z^\bot ) \rangle - Ric_{\mathcal {N}} (X^\bot , Z^\bot ) - |A|^2\langle X^\bot , Z^\bot \rangle \ d\mathscr {H}^n \\&\qquad + \int _{\Sigma } \langle X^{\bot }, H\rangle \langle Z^{\bot },H \rangle - \langle \nabla _{X^{\bot }} (Z^{\bot }), H \rangle - \langle [X^{\bot },Z^{\top }],H \rangle \ d\mathscr {H}^n \\&\qquad + \int _{\partial \Sigma } \langle {\nabla }_{X^{\bot }} (Z^{\bot }), \nu \rangle + \langle [X^{\bot },Z^{\top }],\nu \rangle + div_{\Sigma }(Z^{\top })\langle X^{\top }, \nu \rangle \ d\mathscr {H}^{n-1} \\&\qquad - \int _{\partial \Sigma } \langle Z^{\bot }, H \rangle \langle X^{\top }, \nu \rangle + \langle X^{\bot }, H \rangle \langle Z^{\top }, \nu \rangle \ d\mathscr {H}^{n-1}. \end{aligned}$$

We remark that, in the above formula, whereas X and Z are tangent to $\partial {\mathcal {N}}$ by assumption, $X^{\top }, X^{\bot }, Z^{\top }$ and $Z^{\bot }$ are tangent to $\partial {\mathcal {N}}$ if and only if $\nu$ is orthogonal to $\partial {\mathcal {N}}$.

When $\Sigma$ is a free boundary minimal hypersurface in ${\mathcal {N}}$ we recover the usual second variation formula,

$$\begin{aligned} {\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0} = \check{Q}_{\Sigma }(X^{\bot },Z^{\bot })=\check{Q}_{\Sigma }(Z^{\bot },X^{\bot }). \end{aligned}$$

where $\check{Q}_{\Sigma }$ is the index form on the normal bundle of $\Sigma$. In fact, in that situation the terms containing H are obviously zero and, since all components $X^{\top }$, $X^{\bot }$, $Z^{\top }$, $Z^{\bot }$ are orthogonal to $\nu$ (i.e., they are tangent to $\partial {\mathcal {N}}$), $\langle X^{\bot },\nu \rangle$ vanishes identically and

$$\begin{aligned} \langle [X^{\top },Z^{\bot }] ,\nu \rangle= & {} \left\langle \nabla _{X^{\top }}(Z^{\bot }),\nu \right\rangle - \left\langle \nabla _{Z^{\bot }}(X^{\top }),\nu \right\rangle \\= & {} \langle II^{\partial {\mathcal {N}}}(X^{\top },Z^{\bot }),\nu \rangle - \langle II^{\partial {\mathcal {N}}}(Z^{\bot },X^{\top }),\nu \rangle = 0. \end{aligned}$$

For the sake of the argument in Sect. 6, it is convenient to compare $\nu$ with the unit outward pointing normal vector field to $\partial {\mathcal {N}}$, denoted by $\nu _{e}$. In a succinct way, we write

$$\begin{aligned}&{\frac{\partial }{\partial t}}\left. {\frac{\partial }{\partial s}}\mathscr {H}^n(\Sigma (t,s))\right| _{t=s=0} \\&\quad =\int _{\Sigma } \langle \nabla ^{\bot }(X^\bot ) , \nabla ^{\bot }(Z^\bot ) \rangle - Ric_{\mathcal {N}} (X^\bot , Z^\bot ) - |A|^2\langle X^\bot , Z^\bot \rangle \ d\mathscr {H}^n \\&\qquad + \int _{\partial \Sigma } \langle \nabla _{X^{\perp }} Z^{\perp }, \nu _e \rangle \ d\mathscr {H}^{n-1} \\&\qquad + \int _{\Sigma } \Xi _1(X,Z,H) \ d\mathscr {H}^n + \int _{\partial \Sigma } \Xi _2(X,Z,H,\nu ,\nu _e) \ d\mathscr {H}^{n-1}, \end{aligned}$$

where

$$\begin{aligned} \Xi _1(X,Z,H) = \langle X^{\bot }, H\rangle \langle Z^{\bot },H \rangle - \langle \nabla _{X^{\bot }} (Z^{\bot }), H \rangle - \langle [X^{\bot },Z^{\top }],H \rangle \end{aligned}$$

and

$$\begin{aligned} \Xi _2(X,Z,H,\nu ,\nu _e) =&\left\langle \nabla _{X^{\bot }}Z^{\bot },\nu -\nu _e \right\rangle + \langle [X^{\bot },Z^{\top }],\nu \rangle + div_{\Sigma }(Z^{\top })\langle X^{\top }, \nu \rangle \\&- \langle Z^{\bot }, H \rangle \langle X^{\top }, \nu \rangle - \langle X^{\bot }, H \rangle \langle Z^{\top },\nu \rangle . \end{aligned}$$

$\Xi _1$ is linear in X and Z, is of first order in these entries, and $\Xi _1 \rightarrow 0$ smoothly as $H\rightarrow 0$ smoothly. $\Xi _2$ depends on X and Z similarly, and $\Xi _2 \rightarrow 0$ smoothly as $H\rightarrow 0$ and $\nu \rightarrow \nu _e$ (for that forces $X^{\bot }$, $X^{\top }$, $Z^{\bot }$ and $Z^{\top }$ to become tangent to $\partial {\mathcal {N}}$)). In the case that $\Sigma$ is uniformly and smoothly close to a fixed minimal and free boundary hypersurface, we will have $\Xi _1, \Xi _2$ uniformly small, and $\langle \nabla _{X^{\perp }} Z^{\perp }, \nu _e \rangle$ uniformly close to $\langle II^{\partial {\mathcal {N}}}( (X^{\bot })^{\top _{\partial {\mathcal {N}}}}, (Z^{\bot })^{\top _{\partial {\mathcal {N}}}}),\nu _e\rangle$ (in a smooth sense).

Appendix B: Proof of Proposition 21

Proof of Proposition 21

Here we need to show that if g is close enough to the Euclidean metric, and t, w are also small enough, then one can find an implicitly defined function $u=u(t,g,w)$ so that $\Phi (t,g,w,u(t,g,w))=(0,0,0)$.

The functional $\Phi$ is $C^1$ (see e.g., Appendix of [39]) and its differential can be computed using the result of Proposition 17 in [2], so that

$$\begin{aligned} D_4\Phi (t,\delta ,0,0)[v]=\frac{d}{ds}_{|{s=0}}\Phi (t,\delta ,0,sv) = \left( \Delta _{\delta }v,\frac{\partial v}{\partial \nu _{\delta }},v|_{\Gamma _2}\right) \end{aligned}$$

where $\delta$ stands for the Euclidean metric and $\nu _{\delta }=-\,\frac{\partial }{\partial x^1}$ in the standard Euclidean coordinates we are adopting.

In order to apply the implicit function theorem we need to study the mapping properties of the linear operator

$$\begin{aligned} D_4\Phi (t,\delta ,0,0): \ Y\rightarrow Z_1\times Z_2\times Z_3. \end{aligned}$$

First of all, it follows at once from Theorem 6,I in [29] together with standard Schauder estimates that for any triple $(f_1,f_2,f_3)\in Z_1\times Z_2\times Z_3$ the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{\delta }v=f_1 &{} \text {in} \ S_{\theta }\\ \frac{\partial v}{\partial \nu _{\delta }}=f_2 &{} \text {in} \ \Gamma _1 \\ v=f_3 &{} \text {in} \ \Gamma _2 \end{array}\right. } \end{aligned}$$

(B.1)

admits one (and only one) solution $u\in C(\overline{S_{\theta }})\cap C^{2,\alpha }(\Omega )$ where $\Omega$ is any relatively compact domain in $\overline{S_{\theta }}\setminus (\Gamma _1\cap \Gamma _2)$. Of course, this implies at once that the map above is injective. For what concerns surjectivity, it is then enough to invoke Theorem 1 in [6], which ensures that any bounded solution to (B.1) does in fact belong to $C^{2,\alpha }(\overline{S_{\theta }})$ provided $\alpha <\alpha _0$ for $\alpha _0=\left\{ \frac{\pi }{2\theta }\right\}$ where $\left\{ x\right\} \in [0,1)$ is defined by $\left\{ x\right\} =x-\lfloor x \rfloor$. In fact, it follows from the explicit barrier construction presented in the first part of the proof of Lemma 1 in the same reference, aimed at handling the situation locally around the edge points, that one gains a global Schauder estimate of the form

$$\begin{aligned} \Vert u\Vert _Y\le C\left( \Vert u\Vert _{C^0(\overline{S_{\theta }})} + \Vert f_1\Vert _{Z_1}+\Vert f_2\Vert _{Z_2}+\Vert f_3\Vert _{Z_3}\right) \end{aligned}$$

(B.2)

for any solution $u\in Y$ of (B.1). In this respect, all we need to check is that this can in fact be upgraded to

$$\begin{aligned} \Vert u\Vert _Y\le C'\left( \Vert f_1\Vert _{Z_1}+\Vert f_2\Vert _{Z_2}+\Vert f_3\Vert _{Z_3}\right) \end{aligned}$$

as this patently completes the proof that $D_4\Phi (t,\delta ,0,0): \ Y\rightarrow Z_1\times Z_2\times Z_3$ is indeed a Banach isomorphism. This is rather standard: assuming by contradiction that were not the case, one could find a sequence $\left\{ u_k\right\} \subset Y$ such that, possibly by renormalizing

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert u_k\Vert _Y=1 \\ \Vert \Delta _{\delta }u_k\Vert _{Z_1}+\Vert \frac{\partial u_k}{\partial \nu _{\delta }}\Vert _{Z_2}+\Vert u_k\Vert _{Z_3}\le 1/k \end{array}\right. } \end{aligned}$$

for all $k\ge 1$ and hence, invoking the Arzelá–Ascoli compactness theorem

$$\begin{aligned} u_k\rightarrow u \ \text {in} \ C^{2}(\overline{S_{\theta }}) \end{aligned}$$

for a subsequence which we shall not rename. In particular, one has at the same time

$$\begin{aligned} \Delta _{\delta }u_k \rightarrow \Delta _{\delta }u \ \ \text {in} \ C^{0}(\overline{S_{\theta }}), \ \text {and} \ \Delta _{\delta }u_k\rightarrow 0 \ \text {in} \ C^{0,\alpha }(\overline{S_{\theta }}) \end{aligned}$$

hence $u\in C^{2}(\overline{S_{\theta }})$ must be a solution of the homogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{\delta }u=0 &{} \text {in} \ S_{\theta }\\ \frac{\partial u}{\partial \nu _{\delta }}=0 &{} \text {in} \ \Gamma _1 \\ u=0 &{} \text {in} \ \Gamma _2 \end{array}\right. } \end{aligned}$$

(B.3)

so that we conclude $u=0$ by virtue of the aforementioned result by C. Miranda. But then $u_k\rightarrow 0$ in $C^2(\overline{S_{\theta }})$ and thus inequality (B.2) applied to $u_k$ immediately implies that $\Vert u_k\Vert _Y\rightarrow 0$ as we let $k\rightarrow \infty$, contrary to the fact that each function of the sequence has been rescaled so to have norm one. This contradiction completes the proof. $\square$