Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

Abstract

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$. Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb {S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \le 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$, we obtain a smooth minimizer $f:{\mathbb {S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$, where $H$ is the mean curvature.

Appendix

1.1 Some useful lemmas

In this subsection we state some useful results we need for proving regularity. Lemma 4 is an extension result adapted to the cut-and-paste procedure we use and is proved in [14].

Lemma 5.1

Let $L$ be a 2-dimensional plane in ${\mathbb {R}^n}, x_0\in L$ and $u\in {{\mathrm{C}}}^\infty (U,L^\perp )$, where $U\subset L$ is an open neighborhood of $L\cap \partial B_\rho (x_0)$. Moreover let $|{{\mathrm{D}}}u|\le c$ in $U$. Then there exists a function $w\in {{\mathrm{C}}}^\infty (\overline{B_\rho (x_0)},L^\perp )$ with the following properties:

$$\begin{aligned}&(1)&w=u\quad \text {and}\quad \frac{\partial w}{\partial \nu }=\frac{\partial u}{\partial \nu }\quad \text {on }\partial B_\rho (x_0), \\&(2)&\frac{1}{\rho }||w||_{L^\infty (B_\rho (x_0))}\le c(n)\left( \frac{1}{\rho }||u||_{L^\infty (\partial B_\rho (x_0))}+||{{\mathrm{D}}}u||_{L^\infty (\partial B_\rho (x_0))}\right) , \\&(3)&||{{\mathrm{D}}}w||_{L^\infty (B_\rho (x_0))}\le c(n)||{{\mathrm{D}}}u||_{L^\infty (\partial B_\rho (x_0))}, \\&(4)&\int _{B_\rho (x_0)}|{{\mathrm{D}}}^2w(x)|^2\,{{\mathrm{d}}}x\le c(n)\rho \int _{{{\mathrm{graph}}}u_{|_{\partial B_\rho (x_0)}}}|A(x)|^2\, d\mathcal{H}^1, \end{aligned}$$

where $d\mathcal{H}^1$ is the 1-dimensional Euclidean Hausdorff measure.

The second lemma is a useful selection principle proved in [16].

Lemma 5.2

Let $\delta >0, I\subset {\mathbb {R}}$ a bounded interval and $A_k\subset I, k\in {\mathbb {N}},$ measurable sets with ${{\mathrm{{\mathcal {L}^1}}}}(A_k)\ge \delta$ for all $k$. Then there exists a set $A\subset I$ with ${{\mathrm{{\mathcal {L}^1}}}}(A)\ge \delta$, such that each point $x\in A$ lies in $A_k$ for infinitely many $k$.

The third lemma is a decay result we need to get the power decay for the $L^2$-norm of the second fundamental form in Lemma 3.6.

Lemma 5.3

Let $g:(0,b)\rightarrow [0, +\infty )$ be a bounded function such that

$$\begin{aligned} g\left( x\right) \le \gamma g(2x)+Cx^\alpha \quad \text {for all }x\in \left( 0,\frac{b}{2}\right) , \end{aligned}$$

where $\alpha >0, \gamma \in (0,1)$, and $C\ge 0$ is a constant. Then there exists a $\beta \in (0,1)$ and a constant $C=C(\gamma ,\alpha ,b,||g||_{L^\infty (0,b)})$ such that

$$\begin{aligned} g(x)\le Cx^\beta \quad \text {for all }x\in \left( 0,b\right) . \end{aligned}$$

The last statement is a generalized Poincaré inequality proved in [16].

Lemma 5.4

Let $\mu >0, \delta \in \left( 0,\frac{\mu }{2}\right)$ and $\Omega =B^{{\mathbb {R}^2}}_\mu (0)\backslash E$, where $E\subset {\mathbb {R}^2}$ is measurable with ${{\mathrm{{\mathcal {L}^1}}}}(p_1(E))\le \frac{\mu }{2}$ and ${{\mathrm{{\mathcal {L}^1}}}}(p_2(E))\le \delta$, where $p_1$ is the projection onto the $x$-axis and $p_2$ is the projection onto the $y$-axis. Then for any $f\in C^1(\Omega )$ there exists a point $(x_0,y_0)\in \Omega$ such that

$$\begin{aligned} \int _\Omega \left| f-f(x_0,y_0)\right| ^2\le C\mu ^2\int _\Omega \left| {{\mathrm{D}}}f\right| ^2+C\delta \mu \sup _\Omega |f|^2 \end{aligned}$$

where $C$ is an absolute constant.

1.2 Definitions and properties of generalized $(r,\lambda )$-immersions

Here we recall the definitions and properties of generalized $(r,\lambda )$-immersions $f:{\mathbb {S}}^2\hookrightarrow M\subset {\mathbb {R}^p}$ appearing in [1].

We call a mapping $A:{\mathbb {R}^p}\rightarrow {\mathbb {R}^p}$ an Euclidean isometry, if there is a rotation $R\in SO(p)$ and a translation $T\in {\mathbb {R}^p}$, such that $A(x)=Rx+T$ for all $x\in {\mathbb {R}^p}$.

For a given point $q\in {\mathbb {S}}^2$ and a given 2-plane $E\in G(p,2)$ let $A_{q,E}:{\mathbb {R}^p}\rightarrow {\mathbb {R}^p}$ be an Euclidean isometry which maps the origin to $f(q)$ and the subspace ${\mathbb {R}^2}\times \{0\}\subset {\mathbb {R}^p}$ onto $f(q)+E$.

Let $U^E_{r,q}\subset {\mathbb {S}}^2$ be the $q$-component of the set $(\pi \circ A^{-1}_{q,E}\circ f)^{-1}(B_r)$, where $\pi :{\mathbb {R}^p}\rightarrow {\mathbb {R}^2}$ is the projection on the first two coordinates.

Definition 5.5

An immersion $f:{\mathbb {S}}^2\hookrightarrow M\subset {\mathbb {R}^p}$ is called a generalized $(r,\lambda )$-immersion, if for each point $q\in {\mathbb {S}}^2$ there is an $E=E(q)\in G(p,2)$, such that $A^{-1}_{q,E}\circ f(U^E_{r,q})$ is the graph of a differentiable function $u:B_r\rightarrow ({\mathbb {R}^2})^\perp$ with $\Vert {{\mathrm{D}}}u\Vert _{C^0(B_r)}\le \lambda$.

The set of generalized $(r,\lambda )$-immersions is denoted by $\mathcal{{F}}^1(r,\lambda )$. Moreover let $\mathcal{{F}}^1_V(r,\lambda )$ be the set of all immersions $f\in \mathcal{{F}}^1(r,\lambda )$ such that $\mu _g({\mathbb {S}}^2)\le V$, where $\mu _g$ is the induced area measure.

A continuous function $f:{\mathbb {S}}^2\hookrightarrow M\subset {\mathbb {R}^p}$ is called a $(r,\lambda )$-function, if for each point $q\in {\mathbb {S}}^2$ there is an $E=E(q)\in G(p,2)$, such that $A^{-1}_{q,E}\circ f(U^E_{r,q})$ is the graph of a Lipschitz function $u:B_r\rightarrow ({\mathbb {R}^2})^\perp$ with with Lipschitz constant $\lambda$. The set of $(r,\lambda )$-functions is denoted by $\mathcal{{F}}^0(r,\lambda )$.

Now we recall the Compactness Theorem in [1, Theorem 0.5].

Theorem 5.6

Let $\lambda \le \frac{1}{4}$. Then $\mathcal{{F}}^1_V(r,\lambda )$ is relatively compact in $\mathcal{{F}}^0(r,\lambda )$ in the following sense: Let $f_k:{\mathbb {S}}^2\hookrightarrow M\subset {\mathbb {R}^p}$ be a sequence in $\mathcal{{F}}^1_V(r,\lambda )$. Then, after passing to a subsequence, there exists a function $f\in \mathcal{{F}}^0(r,\lambda )$ and a sequence of diffeomorphisms $\phi _k:{\mathbb {S}}^2\rightarrow {\mathbb {S}}^2$, such that $f_k\circ \phi _k$ is uniformly Lipschitz bounded and converges uniformly to $f$.