On the entropies of hypersurfaces with bounded mean curvature

Abstract

We are interested in the impact of entropies on the geometry of a hypersurface of a Riemannian manifold. In particular, we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold, provided some geometric assumptions are satisfied. This comparison will follow from the existence of an embedded tube around such a hypersurface. Among the consequences of our study of entropies, we point out some new answers to a question of do Carmo on stable Euclidean hypersurfaces of constant mean curvature.

Appendix

We develop here the missing parts of the proof of Theorem 1. In particular we will prove estimate (2) and equality (3).

The main reason for this Appendix is the lack of references for the computations of the mean curvature equation in general Riemannian manifolds, and for the uniform estimates of its coefficients. The mean curvature equation has been extensively studied in Euclidian space, in Space Forms and more recently in the particular case of constant mean curvature surfaces in homogenous spaces (see for instance [38]). A mean curvature equation similar to our Eq. (51) was obtained in Fermi coordinates for minimal surfaces in [10]. We first give a proof of the cheesebox argument, then we compute the mean curvature equation and deduce the desired estimates of its coefficients. This will fill the gaps in the proof of Theorem 1.

1.1 The cheesebox argument

In this paragraph we recall the cheesebox argument and show how it implies estimate (2). We use the same notations as in the proof of Theorem 1.

Recall that the section P is defined as the graph of the function $\phi$ defined at the beginning of the proof of the second step of Theorem 1.

We first study the case where the ambient space $\mathcal N$ is $\mathbb {R}^{n+1}$.

Let p be a point in M and q be a point above p in the section P (see Fig. 3). Consider the following boxes of $\mathcal {N}:$

$$\begin{aligned} C_p(\rho , h) := exp\left( D_p(\rho ) \times ]0,h[\right) \ \mathrm{and}\ C_q(\rho , h) := exp\left( D_q(\rho ) \times ]0,h[\right) . \end{aligned}$$

where exp is the normal exponential map (see (1)) and $D_p(\rho )$ (respectively $D_q(\rho )$) is a geodesic ball of M (respectively P) around p (respectively around q).

Fig. 3

Intersection of the cheeseboxes at $p\in M$ and at $q\in P$ with $\mathbb {R}\nu _p \oplus \mathbb {R}\nu _q$

We choose $\rho$ as in Theorem 1 so that M and $C_p(\rho , h)$ (respectively P and $C_q(\rho , h)$) intersect only at the boundary side $\partial D \times [0,h]$ of $C_p(\rho , h)$ (respectively at the boundary side $\partial D \times [0,h]$ of $C_q(\rho , h)$). The intersection of $C_p(\rho , h) \cup C_q(\rho , h)$ with the plane through p generated by the normal vectors $\nu (p)$ and $\nu (q)$, is the union of two Euclidean rectangles (see Fig. 3).

For a sufficiently small $\rho ,$ we may choose the height of the box to be proportional to the square of its radius: $h = c\rho ^2$ for some constant c depending on the $C^1$-norm of the the second fundamental form A of M. In fact, in a neighborhood of any $x_0\in M$, the hypersurface M is the graph of the height function h defined on a ball $B_{x_{0}}(\rho )$ of the tangent space at $x_0$ of sufficiently small radius $\rho ,$ and such that for any $x\in B_{x_{0}}(\rho )$, $|h(x)| \le C|x|^2$ where C is a uniform constant.

Indeed, let us clarify the relation between the second fundamental form of the aforementioned graph defined by the height function $h(x_1,\ldots , x_n)$ and the Hessian of h. Notice that $h(0)=0$ and $\nabla h (0)=0.$

Using the Einstein’s convention on indices, we have $|A|^{2}=g^{ik}g^{jl}A_{il}A_{kj}=A_{i}^{j}A_{j}^{i},$ and

$$\begin{aligned} A_i^j = \frac{h_{ik}}{W}g^{kj}, \quad \mathrm{with } \ W:= \sqrt{ 1+ f}, \quad f:=|\nabla h|^2\ \mathrm{and}\ g^{ij}:= \delta ^{ij} - \frac{h_i h_j}{W^3}. \end{aligned}$$

Therefore

$$\begin{aligned} A_i^j = \frac{h_{ij}}{W}- \frac{ h_{ki}h_k h_j }{W^3}, \end{aligned}$$

and computing $|A_i^j h_j |$, we get

$$\begin{aligned} |A_i^j h_j |= \left| \frac{f_i}{2W}\left( 1 -\frac{f}{W^2} \right) \right| . \end{aligned}$$

Since A is bounded, there exists a positive constant c such that $|A_i^j | \le \frac{c}{2\sqrt{n}}$, and applying Cauchy-Schwarz’s inequality, we derive

$$\begin{aligned} 2|A_i^j h_j |= \left| \frac{f_i}{(1+f)^{3/2}}\right| \le c \,\sqrt{f}. \end{aligned}$$

A standard comparison between the solutions of the previous differential inequality and the corresponding differential equality, together with the initial condition $\nabla h(0)=0$, shows that the condition $\sum |x_i|^2 \le \rho$ implies $|\nabla h | \le \frac{c\, \rho }{\sqrt{4-(c\,\rho )^2}}$.

Thus, h is $C^1$-uniformly bounded on the ball $B_{x_{0}}(\rho )$ and W is also uniformly bounded. Finally, $h_{ij} \le C$ on $B_{x_{0}}(\rho )$ (where C is a constant depending on c and $\rho$). In conclusion, since $h(0)=0$ and $\nabla h(0) =0$, we have $|h(x)|\le C|x|^2$ on $B_{x_{0}}(\rho )$.

We may also suppose that the function $\phi$ defining the section P satisfies $\phi \le \frac{h}{2}$ (in Fig. 3, $\phi (p):= d_{\mathbb {R}^{n+1}}(p,q)$). Let $\alpha$ be the angle defined by $\tan \alpha = |\nabla \phi |(p)$ (Fig. 3 represents a limit case for which M and P necessarily intersects for any $q'$ such that $d_{\mathbb {R}^{n+1}}(p,q' )\le d_{\mathbb {R}^{n+1}}(p,q)$ or any $\nu (q')$ such that $\left<\nu (p),\nu (q')\right> \le \cos \alpha$).

For a given $\rho$, if $d_{\mathbb {R}^{n+1}}(p,q)$ is small enough then $|\nabla \phi |\le 1$ unless M and P intersects. Thus $\alpha \le \alpha _0= \frac{\pi }{4}.$ Then, elementary plane geometry gives

$$\begin{aligned} \rho \sin \alpha \le \rho \sin \alpha _0 \le h \left( 1 + \cos \alpha _0 \right) + \phi (p) \cos \alpha _0, \end{aligned}$$

thus

$$\begin{aligned} \rho \sin \alpha \le \left( h + \phi \right) \cos \alpha _0 + h \le 3h, \end{aligned}$$

hence

$$\begin{aligned} \frac{|\nabla \phi |(p)}{\sqrt{1+|\nabla \phi |^2(p)}}\le 3c\rho , \end{aligned}$$

which yields $|\nabla \phi |(p) \le 6c\rho$.

Since, by hypothesis, $\phi (p) \le \frac{c}{2}\rho ^2$ and since the inequality holds for any point $p\in \Omega \subset M$, we obtain

$$\begin{aligned} \Vert \phi \Vert _1 := \left( \sup _{p\in \Omega } | \phi | + \sup _{p\in \Omega } |\nabla \phi | \right) \le O(\rho ). \end{aligned}$$

(49)

Consider the general case, where the ambient space $\mathcal N$ is not necessarily Euclidean. Since $\mathcal N$ has bounded curvature, there exists a radius $\rho _0$ depending on the curvature of $\mathcal N,$ such that for each point $p\in \mathcal N$, there is a harmonic coordinate chart $\psi _p^{-1},$ such that $\psi _p: U \left( := B_0^{\mathbb {R}^{n+1}}\left( \rho _0\right) \right) \subset \mathbb {R}^{n+1} \longrightarrow V := \psi _p\left( U\right) \subset {\mathcal N}$, and the pulled-back metric $g_{\mathcal N}$ is $C^{1,\alpha }$-regular, $C^{1,\alpha }$-close to the Euclidean one. The diffeomorphism $\psi _p$ is $C^1$- uniformly bounded in these coordinates. The previous result concerning Euclidean cheeseboxes applies to $\psi _p^{-1}(M)\cap V$ and $\psi _p^{-1}(P)\cap V$ to prove the $C^1$-uniformly boundedness of $\phi \circ \psi _p$ with respect to $p\in M$. Finally since $\psi _p$ is $C^1$-uniformly bounded with respect to p, so is $\phi .$ In conclusion, there exists a radius $\rho _0,$ depending on the curvature of $\mathcal N$ and M,  such that for each point $p\in M,$ there a cheesebox of M around p of radius $\rho _0$ and height $c\rho _0^2$ in harmonic coordinate charts. For details about the theory of harmonic coordinates see for instance the survey [21] and the references therein.

1.2 CMC equation of a section of the normal bundle of M

Notation is the same as in the previous paragraph and Sect. 3. Our purpose is to compute the mean curvature $H_P$ of the section P in a neighborhood of $q\in P$, in terms of local coordinates around $p\in M$. More precisely we will show how to obtain the expansion (3) of $H_P$ in the proof of Theorem 1.

Let $\psi _p$ be a parametrization of a neighborhood $V_p$ of p in $\mathcal {N}$ as given in previous paragraph: $(\psi _p :B^{{\mathbb R}^{n+1}}_0(R)\subset \mathbb {R}^{n}\times \mathbb {R} \longrightarrow V_p \subset \mathcal {N})$ with $\psi (0) = p$ and $\psi ( B^{{\mathbb R}^{n+1}}_0(R) \cap \mathbb {R}^{n}\times \{0\} ) = M\cap V_p$. The local section $P\cap V_p,$ being in a cheesebox, is parametrized by a graph of a function $\phi : \mathbb {R}^{n}\times \{0\} \longrightarrow \mathbb {R}$. Indeed $\psi _p^{-1}(P\cap V_p)$ and $\psi _p^{-1}(M\cap V_p)$ are $C^{1}$-close in the pulled-back metric $g_{\mathcal {N}}$. For simplicity we identify the metric $g_\mathcal {N}$ of $\mathcal {N}$ with its pulled-back $\psi ^*(g_\mathcal {N})$ on the Euclidean ball $B^{{\mathbb R}^{n+1}}_0(R)$. We denote by $\{e_\alpha \}_{\alpha = 1,\ldots , n+1}$ the standard basis of ${\mathbb R}^{n+1}$ and by $\{e_i\}_{i = 1,\ldots , n}$ the standard basis of ${\mathbb R}^{n}\times \{0\}$. With some abuse of notations, we identify $(x,\phi (x))$ with its image $\psi _p(x,\phi (x))$ and derivatives with respect to $e_i$ of a function f will be denoted by $f_{,i}$.

We first choose an adapted frame tangent to P such that the first n vectors $\{ f_i\left( x ,\phi \left( x\right) \right) := e_i + \phi _{,i}\left( x,\phi \left( x\right) \right) e_{n+1}\}_{i=1,\ldots , n}$ are tangent to P at $\left( x,\phi \left( x\right) \right)$ and the last vector is the unit normal field $\nu _P(x,\phi (x))$ to the graph of $\phi .$ In fact $\nu _{P}$ is a unit vector, solution of the system of linear equations given by $\Big \{g_{\mathcal {N}}\left( \nu _P, F_{i}\right) = 0\Big \}_{ i = 1, \cdots , n}.$ An easy computation yields

$$\begin{aligned} \nu ^\alpha := \frac{1}{W}\left( - g^{\alpha i}\phi _{,i} + g^{\alpha \ n+1}\right) . \end{aligned}$$

where $g^{\alpha \beta }$ is the inverse matrix of $g_{\alpha \beta } := g_{\mathcal {N}}\left( e_{\alpha },e_{\beta }\right) , \alpha ,\beta = 1,\cdots n+1$, and $W^2:= g^{kl}\phi _{,k}\phi _{,l}-2g^{n+1 k }\phi _{,k}+g^{n+1 n+1}$ (we use in all subsequent formulas the Einstein summation convention).

We denote by $\tilde{g}_{ij} := g_{\mathcal N}(f_i,f_j)$ the coefficients of the induced metric on $T_qP.$ Therefore, replacing each $f_i$ by its expression in terms of $e_i$, we obtain

$$\begin{aligned} \tilde{g}_{ij} = g_{ij} + g_{n+1 (j} \phi _{,i)} + g_{n+1 n+1} \phi _{,i}\phi _{,j} \end{aligned}$$

(where $g_{n+1 (j} \phi _{,i)} := g_{n+1 j} \phi _{,i} + g_{n+1 i} \phi _{,j}$).

Let us now compute the mean curvature equation for P. We have

$$\begin{aligned} nH_P(q) = - div(\nu _P)(q) = - \tilde{g}^{ij}g_{\mathcal N}(\nabla _{f_i}\nu _P, f_j )(q) = \tilde{g}^{ij}g_{\mathcal N}(\nu _P, \nabla _{f_i} f_j )(q) \end{aligned}$$

(50)

Therefore, for $\alpha ,\beta ,\gamma = 1,\ldots ,n+1$ and $i,j = 1,\ldots , n$ we obtain

$$\begin{aligned} (\nabla _{f_i}{f_j})^\alpha = \Gamma ^\alpha _{ij} + \Gamma ^\alpha _{n+1 (j} \phi _{,i)} + \Gamma ^\alpha _{n+1 n+1} \phi _{,i}\phi _{,j} +\phi _{,ij}\delta ^{\alpha n+1}. \end{aligned}$$

We then compute $g_{\mathcal N}(\nu _P, \nabla _{f_i} f_j )$ and plug into Eq. (50). We obtain

$$\begin{aligned} nH_P W= & {} \tilde{g}^{ij} \left( \phi _{,ij} + \left( \Gamma ^k_{n+1 n+1} \phi _{,k} + \Gamma ^{n+1}_{n+1 n+1} \right) \phi _{,i}\phi _{,j}\right. \nonumber \\&\left. + \Gamma ^k_{n+1 (i}\phi _{,j)}\phi _{,k} + \Gamma ^{n+1}_{n+1 (i} \phi _{,j)} + \Gamma ^k_{ij}\phi _{,k} + \Gamma ^{n+1}_{ij}\right) \end{aligned}$$

(51)

Now, we use harmonic charts, described at the end of Sect. 7.1.

In harmonic charts, the induced metric $g_{\mathcal {N}}$ of $\mathcal {N}$ on $B^{{\mathbb R}^{n+1}}_p(\rho )$ is $C^{1,\alpha }$- regular and $C^{1,\alpha }$ uniformly close to the Euclidean metric. Since $\phi$ is also $C^1$ uniformly bounded (see (49)), the coefficients of Eq. (51) are $C^{0,\alpha }$ uniformly bounded.

By Schauder theory, we obtain uniform $C^\infty$ bounds on $\phi$.

Notice first that, when $\phi =0,$ Eq. (51) gives the mean curvature of the zero section

$$\begin{aligned} nH_M g^{n+1 n+1} = g^{ij} \Gamma ^{n+1}_{ij} \end{aligned}$$

(52)

Replacing Eq. (52) in Eq. (51) we obtain the estimate

$$\begin{aligned} nH_P = nH_M + \tilde{g}^{ij} \phi _{,ij} + O(\rho ^\alpha ) = nH_M + \Delta _M\phi + O(\rho ^\alpha ) \end{aligned}$$

(53)

which gives estimate (3) in the proof of Theorem 1