Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces

Abstract

We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \ge 2$, $p\in (1, \, +\infty )$ and $\Sigma$ an n-dimensional, closed hypersurface in $\mathbb {R}^{n+1}$, which is the boundary of a convex, open set. We show that if the $L^p$-norm of the trace-free part of the anisotropic second fundamental form is small, then $\Sigma$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.

Appendix A: Proof of Computational Propositions

In this section, we prove the computational propositions stated in Sects. 6 and 8. As in Sect. 6, we will differentiate the geometric quantities associated to $\Sigma$ and $\mathcal {W}$. In particular, we denote the normal and the second fundamental form of $\Sigma$ by $\nu ^\Sigma$ and $h^\Sigma$, respectively, while we omit the dependence on $\mathcal {W}$ for the same quantities of the Wulff shape.

Proof of Proposition 6.2

The closeness of u stated in (6.5) is trivial, so let us focus on the closeness of $\nabla u$. We prove that if $\varepsilon$ is small, there exists a constant $C(\mathcal {W})$ such that u must satisfy

$$\begin{aligned} \nabla ^2 u \le C( \omega + \nabla u \otimes \nabla u ) \end{aligned}$$

(A.1)

in the sense of the quadratic forms. In order to prove (A.1), we compute the second fundamental form of $\Sigma$. We firstly exhibit the exact expression for $\nu ^\Sigma$. Let x be in $\Sigma$, and let be a frame for $T_x \mathcal {W}$. We compute the differential $d\psi$ in these coordinates, obtaining

$$\begin{aligned} \nabla _i \psi = z_i + \nabla _i u \, \nu + u \nabla _i \nu . \end{aligned}$$

(A.2)

Now we search for a vector $V = \nu + a^i z_i$ which satisfies the condition $\langle V , \nabla _j \psi \rangle = 0$ for every $j=1,\dots ,n$ and we will recover $\nu ^\Sigma =\frac{V}{|V|}$. We compute

$$\begin{aligned} 0&= \langle \nu ^\Sigma , \, \nabla _j \psi \rangle = \langle \nu + a^i z_i , \, z_j + \nabla _j u \, \nu + u \nabla _j \nu \rangle \\&= \nabla _j u + (\omega + u h)^i_j a_i. \end{aligned}$$

Normalizing, we obtain the expression for $\nu ^\Sigma$.

$$\begin{aligned} \nu ^\Sigma = \frac{\nu - \left( {\omega + u h}\right) ^{-1}[\nabla u]}{|{\nu - \left( {\omega + u h}\right) ^{-1}[\nabla u]}|}. \end{aligned}$$

(A.3)

Using the $C^0$ smallness of u we obtain

$$\begin{aligned} \nu ^\Sigma = \frac{\nu - \nabla u}{\sqrt{1 + |{({{\mathrm{Id}}}+ u h)^{-1}[\nabla u]}|^2}} + \mathcal {R}, \end{aligned}$$

(A.4)

where $\mathcal {R}$ is a combination of product of u and $\nabla u$. We use this expression to compute $h^\Sigma$.

$$\begin{aligned} h^\Sigma _{ij}&= \langle \nabla _i \nu ^\Sigma , \, \nabla _j \psi \rangle \nonumber \\&= \frac{\langle \nabla _i \nu + \nabla _i \nabla u, \, z_j + u \nabla _j \nu + \nabla _j u \nu \rangle + \langle \nabla _i \mathcal {R}, \, \nabla _j \psi \rangle }{\sqrt{1 + |{({{\mathrm{Id}}}+ u h)^{-1}[\nabla u]}|^2}} \nonumber \\&= \frac{h_{ij} - \nabla ^2_{ij} u + u h^2_{ij} - u \langle \nabla _i (\nabla u), \, \nabla _j \nu \rangle + h[\nabla u]_i \, \nabla _j u + \langle \nabla _i \mathcal {R}, \, \nabla _j \psi \rangle }{\sqrt{1 + |{({{\mathrm{Id}}}+ u h)^{-1}[\nabla u]}|^2} }. \end{aligned}$$

(A.5)

We notice that in (A.5) every element in $\nabla \mathcal {R}$ must be either a product of $\nabla u$ and $\nabla u$ or a product of u and $\nabla ^2 u$, and every element is controlled by constants depending only on $\mathcal {W}$. Therefore, since u is small we can absorb the products of u and $\nabla ^2 u$ into $\nabla ^2 u$. Since $\Sigma$ is convex, we know that $h^\Sigma \ge 0$, we easily obtain (A.1). We show how this inequality leads to the result. Consider a point $x_{0} \in \mathcal {W}$, and a unit vector $\xi$ in $T_x \mathcal {W}$ which satisfies $\left( { \nabla u(x_{0}), \, \xi }\right) = - ||{\nabla u}||_{C^0}$. Setting $x_\tau = \exp _{x_{0}}(\tau \xi )$ the lemma follows by the simple equality

$$\begin{aligned} u(x_\tau ) - u(x_{0}) = \left( {\nabla u(x_{0}), \, \tau \xi }\right) + \int _{0}^1 t \int _{0}^1 \nabla ^2 u (\gamma (st))[\dot{\gamma }(st), \, \dot{\gamma }(st)] \, dsdt, \end{aligned}$$

where $\gamma :[0, \, 1] \longrightarrow \mathcal {W}$ is the geodesic which connects $x_{0}$ and $x_\tau$. Applying (A.1) we find

$$\begin{aligned} u(x_\tau ) - u(x_{0})\le & {} \left( {\nabla u(x_{0}), \, \tau \xi }\right) + C\frac{\tau ^2}{2} \left( {1 + ||{\nabla u}||^2_{C^0}}\right) \\= & {} - \tau ||{\nabla u}||_{C^0} + C\frac{\tau ^2}{2} \left( {1 + ||{\nabla u}||^2_{C^0}}\right) . \end{aligned}$$

Finally, for every $\tau$ smaller than the injectivity radius, we obtain the inequality

$$\begin{aligned} ||{\nabla u}||_{C^0} \le \frac{{{\mathrm{osc}}}(u)}{\tau } + \frac{C\tau }{2} \left( {1 + ||{\nabla u}||^2_{C^0}}\right) . \end{aligned}$$

Choosing $\tau = \sqrt{2C {{\mathrm{osc}}}(u) \, (1 + ||{\nabla u}||^2_{C^0})}$ we obtain the result for ${{\mathrm{osc}}}(u)$ small. $\square$

Proof of Proposition 6.3

The proof follows by linearising the metric g of $\Sigma$, its normal $\nu ^\Sigma$ and its second fundamental form $h^\Sigma$. Again, we consider $x \in \Sigma$ and to be a frame for $T_x \mathcal {W}$. We use the differential  A.2 to linearize the metric g.

$$\begin{aligned} g_{ij}&= \langle \nabla _i \psi , \, \nabla _j \psi \rangle = \langle z_i + \nabla _i u \, \nu + u \nabla _i \nu , \, z_j + \nabla _j u \, \nu + u \nabla _j \nu \rangle \\&= \omega _{ij} + 2 u h_{ij} + \nabla _i u \, \nabla _j u + u^2 \underbrace{\langle \nabla _i \nu , \, \nabla _j \nu \rangle }_{= h^k_i h_{kj}} \end{aligned}$$

and we obtain

$$\begin{aligned} |{g_{ij} - \omega _{ij} - 2u h_{ij}}| \le C \sqrt{\varepsilon } \left( {|{u}| + |{\nabla u}|}\right) . \end{aligned}$$

(A.6)

As an easy consequence of (A.6) we obtain the linearisation of the inverse

$$\begin{aligned} |{g^{ij} - \omega ^{ij} - 2u h^{ij}}| \le C \sqrt{\varepsilon } \left( {|{u}| + |{\nabla u}|}\right) \end{aligned}$$

(A.7)

and its determinant

$$\begin{aligned} |{\det g - \det \omega - 2 H u}| \le C \sqrt{\varepsilon } \left( { |{u}| + |{\nabla u}|}\right) . \end{aligned}$$

(A.8)

The linearization of $\nu ^\Sigma$ follows by (A.3). Indeed we find

$$\begin{aligned} \nu ^\Sigma&= \frac{\nu - \left( {\omega + u h}\right) ^{-1}[\nabla u]}{|{\nu - \left( {\omega + u h}\right) ^{-1}[\nabla u]}|} = \nu - \nabla u + \mathcal {R}, \end{aligned}$$

where $\mathcal {R}$ is defined as a combination of product of u and $\nabla u$. We obtain

$$\begin{aligned} |{\nu ^\Sigma - \nu + \nabla u}| \le C \sqrt{\varepsilon } \left( {|{u}| + |{\nabla u}|}\right) . \end{aligned}$$

(A.9)

Now we linearize $h^\Sigma$. We write

$$\begin{aligned} \nu ^{\Sigma } = \nu - \nabla u + \mathcal {R}, \end{aligned}$$

where $\mathcal {R}$ is a combination of products of u and $\nabla u$, we obtain

$$\begin{aligned} h^{\Sigma }_{ij} = \langle \nabla _i \psi , \, \nabla _j \nu ^\Sigma \rangle = \langle z_i + \nabla _i u \, \nu + u \nabla _i, \, \nabla _j \left( {\nu + \nabla u + \mathcal {R}}\right) \rangle \end{aligned}$$

and we obtain

$$\begin{aligned} |{h^\Sigma - h + \nabla ^2 u - h^2 u}| \le C \sqrt{\varepsilon } \left( {|{u}| + |{\nabla u}| + |{\nabla ^2 u }|}\right) . \end{aligned}$$

(A.10)

We prove now (6.7). Let us denote by $\mathcal {R}$ a quantity that can be approximate as

$$\begin{aligned} |{\mathcal {R}}| \le C \left( { |{u}| + |{\nabla u}| + |{\nabla ^2 u}| }\right) . \end{aligned}$$

We obtain

$$\begin{aligned} S_F(\Sigma )&= \left. A_F \right| _{\nu ^\Sigma } h^\Sigma = \left( { \left. A_F \right| _{\nu } - \left. D A_F \right| _{\nu }[\nabla u] }\right) \left( {h - \nabla ^2 u - h^2 u}\right) + \mathcal {R}\\&= \underbrace{\left. A_F \right| _{\nu } h }_{ = S_F(\mathcal {W}) = {{\mathrm{Id}}}} - \underbrace{ \left. D A_F \right| _{\nu }[\nabla u] - A_F \nabla ^2 u }_{\nabla (A_F \nabla u)} - \left. A_F \right| _{\nu } - \underbrace{\left. A_F \right| _{\nu } h^2 u}_{=h u} + \mathcal {R}\end{aligned}$$

and we obtain our estimate. Now we deal with (6.8). Tracking (6.7) we obtain

$$\begin{aligned} |{H_F(\Sigma ) - n + L[u]}| \le C \left( { |{u}| + |{\nabla u}| + |{\nabla ^2 u}| }\right) . \end{aligned}$$

Integrating, we easily obtain

$$\begin{aligned} \left| \overline{H_F(\Sigma )} - n + \fint _\mathcal {W}H u \, dV\right| \le C \sqrt{\varepsilon } \left\| {u}\right\| _{W^{2, \, p}(\mathcal {W})}. \end{aligned}$$

(A.11)

We just have to prove that the integral quantity $\int H u$ is negligible. Here we need the perimeter condition. Indeed, using (A.8) we obtain

$$\begin{aligned} \mathcal {P}(\Sigma ) = \mathcal {P}(\mathcal {W}) + \int _\mathcal {W}H u \, dV + \mathcal {R}, \end{aligned}$$

(A.12)

where $\mathcal {R}$ is again a quantity which can easily be approximated. From (A.12) and the perimeter condition we obtain

$$\begin{aligned} \left| \fint _\mathcal {W}H u \, dV \right| \le C(n,p,F) \sqrt{\varepsilon } ||{u}||_{W^{2, \, p}(\mathcal {W})} \end{aligned}$$

and this concludes the proof. $\square$

Proof of Proposition 8.1

Let $c \in \mathbb {R}^{n+1}$ be given. We easily obtain the estimate

$$\begin{aligned} ||{u - \varphi _u}||_{W^{2, \, p}}&\le ||{u - \varphi _c}||_{W^{2, \, p}} + ||{\varphi _c - \varphi _u}||_{W^{2, \, p}} \le ||{u - \varphi _c}||_{W^{2, \, p}}\\&\quad + \left\| {\sum _{i=1}^{n+1} \langle u - \varphi _u, \, \varphi _i \rangle _{L^2} \varphi _i }\right\| _{W^{2, \, p}} \\&\le ||{u - \varphi _c}||_{W^{2, \, p}} + \sum _{i=1}^{n+1} \left\| { \langle u - \varphi _c, \, \varphi _i \rangle _{L^2} \varphi _i }\right\| _{W^{2, \, p}} \\&\le ||{u - \varphi _c}||_{W^{2, \, p}} + ||{u - \varphi _c}||_{L^1} \sum _{i=1}^{n+1} ||{\varphi _i}||_{L^1} ||{\varphi _i}||_{W^{2, \, p}}\\&\le C(n,p,F) ||{u - \varphi _u}||_{W^{2, \, p}} \end{aligned}$$

and since c is arbitrary, we obtain the thesis. $\square$