Sobolev Inequalities in Spacelike Submanifolds of Minkowski Space

In this paper we are mainly concerned with a specific type of Sobolev inequality for spacelike submanifolds in Minkowski space. Associated to a spacelike submanifold $\Sigma^{m}\hookrightarrow\mathbb{R}^{n,1}$, we define the maximal slope by

$$\tau(\Sigma)=\sup\{|\nu_{0}(x)|:x\in\Sigma,\nu(x)\text{ is a unit normal to }\Sigma\text{ at }x\}.$$

See 2.1 for details. The main results are as follows.

Under the condition of mean convexity, there is no mean curvature term involved, which, to our knowledge, is new, Without the assumption of mean convexity, similar result holds with a curvature term involved.

Here $\|H\|^{2}=-|H|^{2}\geq 0$; see Section 2 for details. Next we establish the same Sobolev inequality for submanifold $\Sigma^{m}\subseteq\mathbb{R}^{n,1}$ of higher codimension $n-m+1$, with $0<m<n$. Let $\nu$ be a normal vector field of $\Sigma$ with $|\nu|^{2}=-1$. We then write $T_{x}^{\bot}\Sigma=T_{x}^{\bot,1}\Sigma\oplus T_{x}^{\bot,2}\Sigma$, where $T_{x}^{\bot,2}\Sigma=\operatorname{Span}\{\nu(x)\}$. Accordingly, the mean curvature vector is decomposed into $H^{\bot,1}+H^{\bot,2}$.

At the end of this work, we discovered that in [TW22] similar results were established. Nevertheless, compared to the results therein, our construction yields a Sobolev inequality without curvature terms for a mean convex hypersurface. The history of geometric inequalities probably dates back to ancient Greece. The classical isoperimetric inequality asserts that for a domain $\Sigma\subseteq\mathbb{R}^{n}$ with sufficiently well-behaved boundary, there holds

(1.4)

$$|\partial\Sigma|\geq n\omega_{n}^{\frac{1}{n}}|\Sigma|^{\frac{n-1}{n}}.$$

It is known that such an isoperimetric inequality is essentially equivalent to the $W^{1,1}$ Sobolev inequality for domains in Euclidean space:

(1.5)

$$\int_{\Sigma}|\nabla f|+\int_{\partial\Sigma}f\geq n\omega_{n}^{\frac{1}{n}}\left(\int_{\Sigma}f^{\frac{n}{n-1}}\right)^{\frac{n-1}{n}}.$$

Intensive work has been established to extend Eq. 1.4 or Eq. 1.5 to more general settings. It has long been conjectured that the same inequality Eq. 1.4 holds in Cartan-Hadamard manifolds [Aub76]. Partial results include [Wei26, BR33, Cro84, Kle92]. See also [GS21] for a recent attempt to resolve the conjecture. It is also possible to replace areas and volumes in Eq. 1.4 by more general quermassintegrals. The resulting isoperimetric inequality is proved for hypersurfaces with certain convexity in Euclidean space. See [Gua] for details.

For a domain $\Sigma$ in a two dimensional space form of constant curvature $K$, there is a neat result which states that

(1.6)

$$4\pi|\Sigma|\leq|\partial\Sigma|^{2}+K|\Sigma|^{2}.$$

Please see [Cho05] and references therein. The same inequality Eq. 1.6 is proved by Choe and Gulliver [CG92a] for minimal surfaces $\Sigma^{2}$ with certain topological constraints in hyperbolic space $\mathbb{H}^{n}$. Yau [Yau75], Choe and Gulliver [CG92] showed that if $\Sigma$ is a domain in $\mathbb{H}^{n}$ or a $n$-dimensional minimal submanifold in $\mathbb{H}^{n+m}$, then it satisfies the linear isoperimetric inequality

$$(n-1)|\Sigma|\leq|\partial\Sigma|.$$

Another linear inequality for proper minimal submanifolds in $\mathbb{H}^{n}$ is obtained in [MS14] using Poincaré model.

It is a longstanding conjecture that Eq. 1.4 holds true for minimal hypersurfaces in $\mathbb{R}^{n+1}$. Using the method of sliding, Brendle fully settled the problem in a recent work [Bre21], and later extended his results to Riemannian manifolds with nonnegative Ricci curvature [Bre20]. The most classical application of the sliding method is perhaps Aleksandrov’s maximum principle. In [Cab08] Cabré first employed the sliding method and gave a simple and elegant proof of Eq. 1.4.

We follow Brendle’s method and apply it to submanifolds in the Minkowski space, and obtain some Michael-Simon-Sobolev type inequalities.

Acknowledgements. The author would like to thank Research Professor Qi-Rui Li for his instructions and many helpful discussions.

Let $\mathbb{R}^{n,1}$ be the Minkowski space endowed with metric

$$\bar{g}=-dx_{0}^{2}+dx_{1}^{2}+\cdots+dx_{n}^{2}.$$

The usual Euclidean metric of $\mathbb{R}^{n+1}$ is denoted by $\delta$. The volume element of $\mathbb{R}^{n,1}$, $d\mu_{\bar{g}}=\sqrt{-\det\bar{g}}dX$, is just the usual Lebesgue measure. A vector $Y$ is called unit if $|Y|^{2}=\sigma(Y)$, where $\sigma(Y)$ is the signature of $Y$, i.e. $\sigma(X)=1,-1,0$ if $X$ is spacelike, timelike, lightlike, respectively. Finally, we define $\|Y\|=\sqrt{\sigma(Y)|Y|^{2}}$.

Let $(\Sigma^{n},g)\hookrightarrow(\mathbb{R}^{n,1},\bar{g})$ be a spacelike hypersuface. We denote by $x$ a point in $\Sigma$, and by $X(x)$ the corresponding position vector in $\mathbb{R}^{n,1}$. Anything with a ‘bar’ is a quantity of the ambient space. Then the second fundamental form is defined by

$$\bar{\nabla}_{Y}Z=\nabla_{Y}Z-h(Y,Z),\quad\forall Y,Z\in\mathscr{X}(\Sigma).$$

Let $\nu(x)$ be a normal to $\Sigma$ at the point $x$. Clearly $\nu(x)$ is also a vector in $\mathbb{R}^{n,1}$. Denote by $\nu_{\alpha}(x)$ the $\alpha$-th coordinate of $\nu(x)$ in $\mathbb{R}^{n,1}$, so that

$$|\nu|^{2}=-\nu_{0}^{2}+\nu_{1}^{2}+\cdots+\nu_{n}^{2}.$$

The quantity $\tau$ characterizes how ‘lightlike’ $\Sigma$ is; for instance, $\tau=1$ if $\Sigma^{n}\Subset\mathbb{R}^{n}$; and $\tau$ is uniquely determined by the diameter if $\Sigma^{n}\Subset\mathbb{H}^{n}$.

Since Eq. 1.1 is homogeneous in $f$, by normalization we may assume that

(3.1)

$$\int_{\partial\Sigma}f=n\int_{\Sigma}f^{\frac{n}{n-1}}-\int_{\Sigma}|\nabla f|.$$

Let $\eta$ be the outward unit normal of $\partial\Sigma$. Consider the following PDE

(3.2)

$$\begin{cases}\operatorname{div}(f\nabla u)=nf^{\frac{n}{n-1}}-|\nabla f|,&\text{ in }\Sigma\setminus\partial\Sigma,\\ \langle\nabla u,\eta\rangle_{\bar{g}}=1,&\text{ along }\partial\Sigma.\end{cases}$$

By our normalization, the equation has a solution $u\in C^{2,\alpha}$. Since $\Sigma$ is mean convex, we may assume that the mean curvature vector $H$ is either zero or timelike pointing to the past. Let $\nu$ be the unit normal and timelike vector field pointing to the past. Now fix $r>0$. For any $x\in\Sigma$ we define $\mathcal{A}_{r}=\cap_{x\in\Sigma}\Lambda_{r}(x)$, where

(3.3)

$\displaystyle\Lambda_{r}(x)=\left\{p\in\mathbb{R}^{n,1}:|(p-X(x))^{\top}|^{2}<r^{2},-r\leq\langle p-X,\|H\|^{-1}H(x)\rangle\leq 0\right\}.$

Note that when $\|H\|=0$, the notation $\|H\|^{-1}H(x)$ simply means the normal $\nu(x)$.

We write $p-X=s\xi+t\nu$, where $\xi$ is a unit tangent vector and $\nu$ is a unit normal vector pointing to the past. The the definition of $\Lambda_{r}(x)$ implies that $-r<s<r,0\leq t\leq r$. Therefore $\Lambda_{r}(x)$ is a parallelogram illustrated as in Fig. 1. We continue and define

	$\displaystyle\mathcal{U}=\left\{(x,y):x\in\Sigma\setminus\partial\Sigma,y\in T^{\bot}_{x}\Sigma,|\nabla u(x)|<1,-1\leq\langle y,\|H\|^{-1}H(x)\rangle\leq 0\right\},$
	$\displaystyle\mathcal{B}_{r}=\left\{(x,y)\in\mathcal{U}:r\nabla^{2}u(x)+r\langle h(x),y\rangle+g(x)\geq 0\right\}.$

Finally, we take the map $\Phi_{r}:T^{\bot}\Sigma\to\mathbb{R}^{n,1}$,

(3.4)

$$\Phi_{r}(x,y)=X(x)+r(\nabla u(x)+y).$$

In the Riemannian or Lorentzian setting, in order for the area formula to be true, the Jacobian of a map should be modified with volume elements; that is,

$$J(\Phi_{r})=|\det(D\Phi_{r})|\cdot\frac{\sqrt{-\det\bar{g}}}{\sqrt{\det g}},$$

where $D\Phi_{r}$ is the usual tangent map of the coordinate map.

Without the ‘mean convexity’ assumption, the union $\mathcal{A}_{r}=\cap_{x\in\Sigma}\Lambda_{r}(x)$, with $\Lambda_{r}(x)$ defined by Eq. 3.3, might as well be empty. We therefore construct $u$ by

(4.1)

$$\begin{cases}\operatorname{div}(f\nabla u)=nf^{\frac{n}{n-1}}-\sqrt{|\nabla f|^{2}+f^{2}\|H\|^{2}},&\text{ in }\Sigma,\\ \langle\nabla u,\eta\rangle_{\bar{g}}=1,&\text{ along }\partial\Sigma.\end{cases}$$

Since Eq. 1.2 is homogeneous in $f$, we may assume

(4.2)

$$\int_{\partial\Sigma}f=\int_{\Sigma}nf^{\frac{n}{n-1}}-\int_{\Sigma}\sqrt{|\nabla f|^{2}+f^{2}\|H\|^{2}},$$

so that Eq. 4.1 admits a solution. We then modify Eq. 3.3 by

$$\Lambda_{r}(x)=\left\{p\in\mathbb{R}^{n,1}:|(p-X(x))^{\top}|^{2}<\frac{r^{2}}{4},-\frac{r}{2}\leq\langle p-X(x),\|H\|^{-1}H(x)\rangle\leq\frac{r}{2}\right\}$$

and $\mathcal{A}_{r}=\cap_{x\in\Sigma}\Lambda_{r}(x)$. Accordingly,

	$\displaystyle\mathcal{U}=\left\{(x,y):x\in\Sigma\setminus\partial\Sigma,y\in T^{\bot}_{x}\Sigma,|\nabla u(x)|<\frac{1}{2},-\frac{1}{2}\leq\langle y,\|H\|^{-1}H(x)\rangle\leq\frac{1}{2}\right\},$
	$\displaystyle\mathcal{B}_{r}=\left\{(x,y)\in\mathcal{U}:r\nabla^{2}u(x)+r\langle h(x),y\rangle+g(x)\geq 0\right\},$

and $\Phi_{r}(x,y)=X(x)+r(\nabla u(x)+y)$. Note that when $\|H\|=0$, the notation $\|H\|^{-1}H$ simply means the normal vector $\nu$. Similar as in Section 3, we still have

• •

  The inclusion $\Phi_{r}(\mathcal{B}_{r})\supseteq\mathcal{A}_{r}$; and

• •

  The Jacobian $J(\Phi_{r})=r^{n+1}\det\left(\nabla_{i}\nabla^{j}u(x)+\langle y,h_{i}^{j}\rangle+\delta_{i}^{j}/r\right)$.

The proofs are almost identical.

We fix such a section $\nu\in\Gamma(T^{\bot}\Sigma)$, with $|\nu|^{2}=-1$; that is, $\nu$ is a unit timelike normal vector field along $\Sigma$. At each point $x\in\Sigma$, we have decompostion

$$T_{x}^{\bot}\Sigma=T_{x}^{\bot,1}\Sigma\oplus T_{x}^{\bot,2}\Sigma,$$

where $T_{x}^{\bot,2}\Sigma=\operatorname{Span}\{\nu(x)\}$. Note that $T_{x}^{\bot,1}\Sigma$ is a spacelike subspace. Accordingly, we write a normal vector as $y=y^{\bot,1}+y^{\bot,2}$. The equation we consider now is

(5.1)

$$\begin{cases}\operatorname{div}(f\nabla u)=mf^{\frac{m}{m-1}}-\sqrt{|\nabla f|^{2}+f^{2}(|H^{\bot,1}|^{2}+\|H^{\bot,2}\|^{2})},&\Sigma\setminus\partial\Sigma,\\ \langle\nabla u,\eta\rangle=1,&\partial\Sigma.\end{cases}$$

We normalize by

(5.2)

$$\int_{\partial\Sigma}f=\int_{\Sigma}mf^{\frac{m}{m-1}}-\int_{\Sigma}\sqrt{|\nabla f|^{2}+f^{2}(|H^{\bot,1}|^{2}+\|H^{\bot,2}\|^{2})}$$

so that the PDE has a solution. The domains and codomains are

	$\displaystyle\Lambda_{r}(x)=\left\{p\in\mathbb{R}^{n,1}:|(p-X(x))^{\top}|^{2}+|(p-X(x))^{\bot,1}|^{2}<\frac{r^{2}}{4},-\frac{r}{2}\leq\langle p-X(x),\nu(x)\rangle\leq\frac{r}{2}\right\},$
	$\displaystyle\mathcal{A}_{r}=\cap_{x\in\Sigma}\Lambda_{r}(x),$
	$\displaystyle\mathcal{U}=\left\{(x,y):x\in\Sigma\setminus\partial\Sigma,y\in T_{x}^{\bot}\Sigma,|\nabla u(x)|^{2}+|y^{\bot,1}|^{2}<\frac{1}{4},-\frac{1}{2}\leq\langle y^{\bot,2},\nu\rangle\leq\frac{1}{2}\right\},$
	$\displaystyle\mathcal{B}_{r}=\{(x,y)\in\mathcal{U}:r\nabla^{2}u(x)+r\langle y,h(x)+g(x)\geq 0\rangle\}.$

Finally we take $\Phi_{r}:T^{\bot}\Sigma\to\mathbb{R}^{n,1}:(x,y)\mapsto X(x)+r(\nabla u(x)+y)$. As in Section 3,

• •

  The inclusion $\Phi_{r}(\mathcal{B}_{r})\supseteq\mathcal{A}_{r}$; and

• •

  The Jacobian $J(\Phi_{r})=r^{n+1}\det\left(\nabla_{i}\nabla^{j}u(x)+\langle y,h_{i}^{j}\rangle+\delta_{i}^{j}/r\right)$.

The proofs are almost identical.