The existence of embedded G-invariant minimal hypersurface

Abstract

For a compact connected Lie group G acting as isometries of cohomogeneity not equal to 0 or 2 on a compact orientable Riemannian manifold $M^{n+1},$ we prove the existence of a nontrivial embedded G-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.$

Appendix A. Appendix

1.1 A.1. Ball covering of tubes

We will prove the following useful lemma.

Lemma A.1

For any $y\in M,$ there exists $\rho _0>0$ so that for any $\rho <\rho _0,$ there exists a collection ${\mathcal {B}}$ of disjoint geodesic balls of radius $\rho$ with centers in G.y so that the concentric balls with radius $5\rho$ covers $B^G_\rho (y)$. Moreover, the number of balls in this collection is at most $C_y \rho ^{-d_y},$ where $C_y$ is a constant depending only on G.y,  and $d_y=\dim G.y.$

Proof

Here we use a basic 5-times-radius covering theorem (putting $\tau =2$ and $\delta$ as diameter in 2.8.5 in [6]), that says for a covering using metric balls in metric space, we can find a disjoint subcollection so that 5-times-radius concentric balls of this subcollection would cover all the original balls. Now consider the covering of $B_{\rho }^G(y)$ by $\{B_{\rho }(z)|z\in G.y\}.$ We deduce that there exists a set ${\mathcal {B}}$ consisting of finitely many points so that $B_{\rho }(z)\cap B_{\rho }(z')=\varnothing$ if $z\not =z',z,z'\in {\mathcal {B}}$ and

$$\begin{aligned} B^G_{\rho }(y)\subset \bigcup _{z\in {\mathcal {B}}}B_{5\rho }(z). \end{aligned}$$

Note that the cardinality of ${\mathcal {B}}$ satisfies the following obvious bound

$$\begin{aligned} |{\mathcal {B}}|\le \frac{\text {Vol}(B_{\rho }^G(y))}{B_{\rho }(y)}, \end{aligned}$$

since G acts by isometries and thus pushes forward geodesic balls to geodesic balls.

Let

$$\begin{aligned} d_y=\dim G.y. \end{aligned}$$

Recall the volume of tubes in [8]. There exists $\rho _0>0$ so that for all $\rho <\rho _0$, we would have

$$\begin{aligned} \text {Vol}(B^G_{\rho }(y))\le C_{d_y} {\mathcal {H}}^{d_y}(G.y)\rho ^{n+1-d_y}, \end{aligned}$$

for some dimensional constant $C_{d_y}>0.$ Moreover, by the volume of geodesic balls, we could assume that $\text {Vol}(B_{\rho }(y))\ge C_{n}\rho ^{n+1},$ for some dimensional constant $C_n>0$ by shrinking $\rho _0$ if necessary. If $\rho <\rho _0,$ then there exists $C_y>0$ depending only on G.y and M such that

$$\begin{aligned} |{\mathcal {B}}|\le C_y \rho ^{-d_y}. \end{aligned}$$

$\square$

1.2 Splitting of tangent cone of integral G-varifold

Let G.x be an orbit of dimension $d_x$, and $B_\rho ^G(x)$ be the $\rho$-tubular neighborhood around x. Suppose V is a rectifiable G-varifold in ${\mathbf {V}}_n$ and x is in $\text {spt}V$. We will prove the following lemma which implies that the tangent cone splits as a product into normal directions and tangential directions to G.x.

Lemma A.2

For any point $y\in G.x,$ there exists a tangent cone $C_y\subset T_y M$ of V, so that $C_y+w=C_y$ for any $w\in T_y G.x\subset T_y M.$

Proof

Without loss of generality, we can assume $y=x,$ since we can always pushforward our constructions by any element of g. We will use $\exp$ to denote the restriction of exponential map in $T_yM$ inside a ball of injectivity radius. Let $r_i\rightarrow \infty$ be a sequence so that $(r_i )_{\#}\exp ^{-1}_{\#}(V)\rightarrow C$ as varifold, where $r_i$ is multiplication by $r_i$ in $T_yM.$ Note that we have

$$\begin{aligned} (r_i)_{\#}\exp ^{-1}_{\#}G.y=i_{\#}T_y G.y, \end{aligned}$$

if i is the inclusion $G.y\hookrightarrow M.$

By Main Theorem of [15], we can isometrically embed M into some ${\mathbb {R}}^N$ so that the action of G on M comes from a linear representation of G on ${\mathbb {R}}^N.$ We will also denote this action as $\rho (g)z$ for $z\in {\mathbb {R}}^N.$ We will identify M as a submanifold of ${\mathbb {R}}^N$ in the following reasoning.

Let $c\in C_y$ be a point in the tangent cone. We will also regard it as a vector. We can find a sequence of points $c_j\in V$ so that $r_j\exp _{\#}^{-1}c_j\rightarrow c.$ Let g(t) be a smooth path in G so that $g(0)=0$,

$$\frac{d}{dt}\bigg |_{t=0}g(t).y=w.$$

Such a path exists by lifting a corresponding path starting with $w\in T_y G.y$ and staying in $G.y\approx G/G_y$.

Now, note that $g(r_i^{-1}).c_i\in V.$ If we can prove that

$$\begin{aligned} r_i\exp ^{-1}_{\#}(g(r_i^{-1}).c_i)=c+w, \end{aligned}$$

(A.1)

then we are done. To prove this, we need to compare $\exp _{\#}^{-1}(z)$ with $z-y$. First, note that $d(z-y)|_{T_y{\mathbb {R}}^N}=\text {id}_{T_y{\mathbb {R}}^N}=d\exp _{\#}^{-1}(z)|_{T_y{\mathbb {R}}^N}.$ Thus, $\exp _{\#}^{-1}(z)-(z-y)=O(d_M(z,y)^2)$ for $z\in M.$. Thus, we have

$$\begin{aligned}&r_i\exp ^{-1}_{\#}(g(r_i^{-1}).c_i)\\&\quad =r_i(g(r_i^{-1}).c_i-y)+r_iO(d_M(g(r_i^{-1}).c_i,y)^2)\\&\quad =r_i(\rho (g(r_i^{-1}))c_i-\rho (g(r_i^{-1}))y+\rho (g(r_i^{-1}))y-y)+r_iO(\left\Vert \exp ^{-1}_{\#}(g(r_i^{-1}).c_i)\right\Vert ^2)\\&\quad =\rho (g(r_i^{-1}))r_i\exp ^{-1}(c_i)+r_iO(\left\Vert \exp ^{-1}(g(r_i^{-1}).c_i)\right\Vert ^2)\\&\qquad +r_i(\rho (g(r_i^{-1}))-\rho (g(0)))y+r_iO(\left\Vert \exp ^{-1}g(r_i^{-1})y\right\Vert )+r_iO(\left\Vert \exp ^{-1}_{\#}(g(r_i^{-1}).c_i)\right\Vert ^2)\\&\quad =\rho (g(r_i^{-1}))r_i\exp ^{-1}(c_i)+r_i(\rho (g(r_i^{-1}))-\rho (g(0)))y+O(r_i^{-1}). \end{aligned}$$

Let $i\rightarrow \infty$, and we immediately get A.2. $\square$

Corollary 3

The support of any such cone $C_y$ as in A.2 is a product of $T_yG.x$ and a rectifiable set W supported in $i_{\#}(T_yG.x)^\perp .$

Proof

Take $W=C_y\cap i_{\#}(T_yG.x)^\perp$ and use Lemma A.2. $\square$