Embedded constant mean curvature surfaces in Euclidean three-space

Abstract

In this paper we refine the construction and related estimates for complete constant mean curvature surfaces in Euclidean three-space developed in Kapouleas (Ann Math 131:239–330, 1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (Invent Math 119(3):443–518, 1995). As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in Kapouleas (Ann Math 131:239–330, 1990) and we produce a very large class of new embedded examples of finite topology.

Appendices

Appendix A: Delaunay surfaces

While descriptions of Delaunay surfaces are readily available in the literature, our construction will require an understanding of our choice of conformal immersion into $\mathbb R^3$. To that end, we present, albeit in an abbreviated form, the conformal map and fundamental geometric quantities related to it.

For $\tau \in (0,1/4]$ define the immersion $Y_\tau : \mathbb R\times \mathbb {S}^1 \rightarrow \mathbb R^3$

$$\begin{aligned} Y_\tau (t,\theta ) = \left( k(t), r(t) \cos \theta , r(t) \sin \theta \right) \end{aligned}$$

(8.1)

where

$$\begin{aligned}&r(t)= \sqrt{\tau } e^{w(t)},\\&\left\{ \begin{array}{l} k'(t)= 2\sqrt{\tau }r(t) \cosh w(t)\\ k(0)=0, \end{array} \right. \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{l} ({w'})^2 +4\tau \cosh ^2 w = 1 \\ w(0)>0,\, w'(0)=0.\end{array}\right. \end{aligned}$$

(8.2)

As a result, for any $\tau \in (0,1/4]$, we have Gauss map, metric, and second fundamental form

$$\begin{aligned} \nu _\tau (t,\theta )&=\left( -w', 2\sqrt{\tau }\cosh w \cos \theta , 2\sqrt{\tau }\cosh w \sin \theta \right) ,\end{aligned}$$

(8.3)

$$\begin{aligned} g_\tau&= \tau e^{2w}(dt^2 + d\theta ^2),\end{aligned}$$

(8.4)

$$\begin{aligned} A_\tau&= -2\tau e^w(\sinh w\, dt^2 + \cosh w\, d\theta ^2). \end{aligned}$$

(8.5)

Notice that for $\tau <0$ we replace each $\cosh w$ with a $\sinh w$ and each $\tau$ with $|\tau |$.

With this definition, we see $H \equiv 1$ and

$$\begin{aligned} |A_{\tau }|^2 = 2+2e^{-4w}, \quad K_{\tau }=1-e^{-4w}. \end{aligned}$$

(8.6)

Note that the sign on the curvature of the surface corresponds precisely with the sign of $w$.

By the nature of the equation and based on its initial conditions, $w$ is periodic with period we designate $2\underline{\mathbf p}_\tau$. Moreover, $w$ has even symmetry about $t=0$ and odd symmetry about $t=\underline{\mathbf p}_\tau /2$. It has a maximum at $t=0$ and a minimum at $t=\underline{\mathbf p}_\tau$. The periodicity of $w$ is preserved in the image surface, and we let $1+\hat{\mathbf p}_\tau =k(\underline{\mathbf p}_\tau )$. Thus, the period of the image surface is $2+2\hat{\mathbf p}_\tau$.

Lemma 8.1

A Delaunay surface with parameter $\tau$, as described above, is rotationally symmetric about the $x_1$-axis and is embedded for $\tau \in (0,\frac{1}{4}]$. Let $r_\tau ^{\mathrm {max}}, r_\tau ^{\mathrm {min}}$ denote the largest and smallest radii of the circles in the $x_2x_3$-plane. Then

$$\begin{aligned} r_\tau ^{\mathrm {max}} = 1+O(\tau ); \, \, r_\tau ^{\mathrm {min}} -(r_\tau ^{\mathrm {min}})^2=\tau \text { for } \tau >0; \,\, r_\tau ^{\mathrm {min}} + (r_\tau ^{\mathrm {min}})^2 = -\tau \text { for } \tau < 0. \end{aligned}$$

Proof

The rotational symmetry and embeddedness follow immediately from the definition of the immersion and fact that $k'>0$ for $\tau \in (0,1/4]$. The ODE for $w$ and initial conditions imply that for $\tau >0$ we have $w(0) = \mathrm {arccosh} \, \left( \frac{1}{2\sqrt{\tau }}\right)$ and $w(\underline{\mathbf p}_\tau )=-\mathrm {arccosh} \, \left( \frac{1}{2\sqrt{\tau }}\right)$. Using the log formulation for $\mathrm {arccosh} \,$,

$$\begin{aligned} r_\tau ^{\mathrm {max}}&= r_\tau (0) =\sqrt{|\tau |}e^{w(0)} = \sqrt{|\tau |} \frac{1}{2\sqrt{|\tau |}}(1+ \sqrt{1- 4\tau })=\frac{1}{2}(1+ \sqrt{1- 4\tau })\\&= 1+O(\tau ). \end{aligned}$$

Note that when $\tau <0$ we replaced $\mathrm {arccosh} \,$ by $\mathrm {arcsinh} \,$ but need not change the formulation as $-\tau = |\tau |$ when $\tau <0$. Observe also that

$$\begin{aligned} r_\tau ^{\mathrm {min}}= r_\tau (\underline{\mathbf p}_\tau ) = \sqrt{|\tau |}e^{w(\underline{\mathbf p}_\tau )} = \sqrt{|\tau |}(2 \sqrt{|\tau |}) (1+\sqrt{1-4\tau })^{-1}. \end{aligned}$$

A quick calculation proves the desired equalities.

The construction relies on the fact that certain regions of Delaunay immersions possess well understood geometric limits as $\tau \rightarrow 0$. We solve the linearized problem with respect to a conformal metric $h$, which behaves on these regions much like the pull back of the Gauss map. In fact, $\nu _\tau ^*g$ provides an isometry between the regions $[-b,b]\times \mathbb S^1$ and $[\underline{\mathbf p}_\tau -b, \underline{\mathbf p}_\tau +b]\times \mathbb S^1$ via the map $Y_\tau (t,\theta ) \rightarrow Y_\tau (\underline{\mathbf p}_\tau -t,\theta )$. (Here $b$ is a large constant, fixed in (5.3).) Therefore, it suffices to understand the asymptotics of the immersion of $[-b,b]\times \mathbb S^1$ by $Y_\tau$.

In [11], the regions and their geometric limits are described in some detail. The next lemma will be stated without proof. The interested reader should consult Lemmas 2.1 and 2.2 in Appendix A of [11] for the details.

Lemma 8.2

Let $r_\tau :[-x_\tau , x_\tau ] \rightarrow \mathbb R$ be the function whose graph, rotated about the $x_1$-axis, gives $Y_\tau ([-\underline{\mathbf p}_\tau , \underline{\mathbf p}_\tau ]\times \mathbb S^1)$. Then as $\tau \rightarrow 0$

$$\begin{aligned} \hat{\mathbf p}_\tau \rightarrow 0, \quad x_\tau \rightarrow 1. \end{aligned}$$

Let $r_0(x_1):[-1,1] \rightarrow \mathbb R$ be defined by $r_0(x_1) = \sqrt{1-x_1^2}$. Given $\epsilon >0$, there exists $\tau _\epsilon >0$ such that if $0<|\tau |< \tau _\epsilon$, then $r_\tau$ restricted to $[-1+\epsilon , 1-\epsilon ]$ depends smoothly on $\tau$ and

$$\begin{aligned} \Vert r_0-r_\tau :C^k([-1+ \epsilon , 1- \epsilon ])\Vert \le C(\epsilon ,k)|\tau |. \end{aligned}$$

Moreover, we have the following period limits as $\tau \rightarrow 0$:

$$\begin{aligned} \lim _{\tau \rightarrow 0} \frac{1}{-\log |\tau |}\cdot \frac{d\hat{\mathbf p}_\tau }{d\tau }=1; \, \, \lim _{\tau \rightarrow 0} \frac{\hat{\mathbf p}_\tau }{-|\tau | \log |\tau |} = 1. \end{aligned}$$

(8.7)

We have the following corollary comparing the metric on the sphere and the Delaunay immersion.

Corollary 8.3

For $\epsilon \in (0,1), k \in \mathbb Z^+$, there exists $\tau _\epsilon >0$ such that for all $0<|\tau |<\tau _\epsilon$,

$$\begin{aligned} \Vert g_\tau - g_0:C^k([-1+\epsilon , 1-\epsilon ]\times \mathbb S^1),g_\tau )\Vert \le C(\epsilon ,k)|\tau |. \end{aligned}$$

(8.8)

For a fixed large $b$ (choose for example the largest $b$ such that $\tanh b = 1-\epsilon$), Lemma 8.2 expresses the limit, as $\tau \rightarrow 0$, of the immersions of the regions $[2\underline{\mathbf p}_\tau n -b,2\underline{\mathbf p}_\tau n + b] \times \mathbb S^1$ for each $n \in \mathbb Z$. Regions of the form $[(2n-1)\underline{\mathbf p}_\tau -b, (2n-1)\underline{\mathbf p}_\tau +b] \times \mathbb S^1$ are isometric to these regions in the metric $\nu _\tau ^*g_\tau$ under the mapping $Y_\tau (t,\theta ) \rightarrow Y_\tau (\underline{\mathbf p}_\tau -t,\theta )$. On regions in between, one cannot appeal to natural geometric limiting behavior. Instead, we understand the behavior of these portions of the cylinder in the flat metric $dt^2 + d\theta ^2$. We determine the limiting length of such a cylindrical piece in this metric.

Lemma 8.4

$$\begin{aligned} \underline{\mathbf p}_\tau =-\frac{1}{2}\log \tau + O(1); \quad \frac{d \underline{\mathbf p}_\tau }{d\tau }= - \frac{1}{2\tau }+O(1). \end{aligned}$$

(8.9)

Proof

As $w'$ has a sign on $[0, \underline{\mathbf p}_\tau ]$ we may invert and compute

$$\begin{aligned} 0-\underline{\mathbf p}_\tau = \int _{w(\underline{\mathbf p}_\tau )}^{w(0)} \frac{dt}{dw} dw = -\int _{\mathrm {arccosh} \, (\frac{1}{2\sqrt{\tau }})}^0 \frac{dw}{\sqrt{1-4\tau \cosh ^2 w}}. \end{aligned}$$

Making the substitution $u=e^w$, and letting $a_\tau := \exp (\mathrm {arccosh} \, (\frac{1}{2\sqrt{\tau }}))=\frac{1}{2\sqrt{\tau }}\left( 1+ \sqrt{1-4\tau }\right)$ we see

$$\begin{aligned} \underline{\mathbf p}_\tau = \int _{a_\tau }^1 \frac{du}{\sqrt{u^2-\tau (u^2+1)^2}}. \end{aligned}$$

Substituting again, this time with $v = u/(u^2+1)$ we get

$$\begin{aligned} \underline{\mathbf p}_\tau = \int _{\sqrt{\tau }}^{1/2} \frac{dv}{\sqrt{v^2-\tau }}=\hbox {log}\left( \sqrt{\frac{1}{4}-\tau }+\frac{1}{2}\right) -\frac{1}{2}\hbox {log }\tau . \end{aligned}$$

Explicit calculation then gives the result.

Appendix B: Quadratic estimates

For completeness, we include here a proposition we will need, the proof of which appears in the appendix of [21] (see also [7]). Let $X:D \rightarrow U$ be an immersion of the unit disk in $\mathbb R^2$ into an open cube $U\subset \mathbb R^3$ equipped with a metric $g$. Assume $\mathrm {dist}_g(X(D), \partial U)>1$ and there exists $c_1>0$ such that:

$$\begin{aligned} \Vert \partial X: C^{2,\beta }(D,g_0)\Vert \le c_1, \,\,\,\Vert g_{ij}, g^{ij}:C^{4, \beta }(U,g_0)\Vert \le c_1, \,\,\,g_0 \le c_1X^*g, \end{aligned}$$

(9.1)

where $\partial X$ represents the partial derivatives of the coordinates of $X$, $g^{ij}$ are the components of the inverse of the metric $g$, and $g_0$ denotes the standard Euclidean metric on $D$ or $U$ respectively. We note that (9.1) can be arranged by an appropriate magnification of the target, which we will exploit in order to make use of the following proposition.

Let $\nu :D \rightarrow \mathbb R^3$ be the unit normal for the immersion $X$ in the $g$ metric. Given a function $\phi :D \rightarrow \mathbb R$ which is sufficiently small, we define $X_\phi :D \rightarrow U$ by

$$\begin{aligned} X_\phi (p):= \exp _{X(p)}(\phi (p)\nu (p)) \end{aligned}$$

(9.2)

where $\exp$ is the exponential map with respect to the $g$ metric. Then the following holds:

Proposition 9.1

There exists a constant $\epsilon (c_1)>0$ such that if $X$ is an immersion satisfying (9.1) and the function $\phi :D \rightarrow \mathbb R$ satisfies

$$\begin{aligned} \Vert \phi :C^{2,\beta }(D,g_0)\Vert \le \epsilon (c_1) \end{aligned}$$

then $X_\phi :D \rightarrow U$ is a well defined immersion by (9.2) and satisfies

$$\begin{aligned} \Vert X_\phi - X - \phi \nu :C^{1,\beta }(D,g_0)\Vert \le C(c_1)\Vert \phi :C^{2,\beta }(D,g_0)\Vert ^2 \end{aligned}$$

and

$$\begin{aligned} \Vert H_\phi - H - \mathcal L_{X^*g} \phi :C^{0, \beta }(D,g_0)\Vert \le C(c_1)\Vert \phi : C^{2, \beta }(D,g_0)\Vert ^2. \end{aligned}$$

Here $H=tr_g A$ is the mean curvature of $X$, defined with respect to the metric $X^*g$ where $A$ is the second fundamental form, $H_\phi$ the mean curvature of $X_\phi$, and $\mathcal L_{X^*g}:= \varDelta _g + |A|^2$.