Embedded loops in the hyperbolic plane with prescribed, almost constant curvature

Abstract

Given a constant $k>1$ and a real-valued function K on the hyperbolic plane ${\mathbb {H}}^2$, we study the problem of finding, for any $\varepsilon \approx 0$, a closed and embedded curve $u^\varepsilon$ in ${\mathbb {H}}^2$ having geodesic curvature $k+\varepsilon K(u^\varepsilon )$ at each point.

Loops in the Euclidean plane

The argument we used to prove Theorem 1.1 applies also in the easier Euclidean case. It is well known that the only embedded loops in ${\mathbb {R}}^2$ having prescribed constant curvature $k>0$ are circles of radius 1 / k. We take as a reference circle the loop

$$\begin{aligned} \omega (x)=\frac{1}{k} ~x~,\qquad x\in {\mathbb {S}}^1\subset {\mathbb {R}}^2, \end{aligned}$$

which solves

$$\begin{aligned} u''=L(u) k~iu'~,\quad \text {where}\quad L(u):=\Big ({\mathop {\fint }\limits _{{\mathbb {S}}^1}}|u'|^2~\hbox {d}x\Big )^\frac{1}{2} \end{aligned}$$

(in fact, $L(\omega )k=1$ and $\omega ''=-\omega =i\omega '$).

Let $K\in C^1({\mathbb {R}}^2)$ be given. If a nonconstant function $u\in C^2({\mathbb {S}}^1,{\mathbb {R}}^2)$ solves

$$\begin{aligned} u''=L(u)(k+\varepsilon K(u))~iu'~, \end{aligned}$$

(A.1)

then $|u'|=L(u)$ is constant, and u parameterizes a loop in ${\mathbb {R}}^2$ having Euclidean curvature $k+\varepsilon K$ at each point. Further, problem (A.1) admits a variational structure, see [5, 18]. More precisely, its nonconstant solutions are critical points of the energy functional

$$\begin{aligned} E_{k+\varepsilon K}(u)=\Big ({\mathop {\fint }\limits _{{\mathbb {S}}^1}}|u'|^2~\hbox {d}x\Big )^\frac{1}{2}+\varepsilon {\mathop {\fint }\limits _{{\mathbb {S}}^1}}Q(u)\cdot iu'~\hbox {d}x~,\quad u\in C^2({\mathbb {S}}^1,{\mathbb {R}}^2){\setminus }{\mathbb {R}}^2, \end{aligned}$$

where the vector field $Q\in C^1({\mathbb {R}}^2,{\mathbb {R}}^2)$ satisfies ${\mathrm{div}}Q=K$.

Arguing as for Theorem 4.1, one can prove a necessary conditions for the existence of solutions to (A.1) for $\varepsilon =\varepsilon _h\rightarrow 0$.

Theorem A.1

Let $u_h$ be a $(k+\varepsilon _hK)$-loop solving (A.1) for $\varepsilon =\varepsilon _h$, and assume that

$$\begin{aligned} L(u_h)\rightarrow L_\infty >0, \qquad u_h\rightarrow U\hbox { uniformly, for some }U\in C^0({\mathbb {S}}^1,{\mathbb {R}}^2). \end{aligned}$$

Then, $U(x)=\omega \big (\xi x^\mu )+z$ for some $\mu \in {\mathbb {N}}$, $\xi \in {\mathbb {S}}^1$ and $z\in {\mathbb {R}}^2$ that is a critical point for the Melnikov function

$$\begin{aligned} F^{K}_k(z)=\int \limits _{D_{\frac{1}{k}}(z)}K(q)~\hbox {d}q~,\quad F^{K}_k:{\mathbb {R}}^2\rightarrow {\mathbb {R}}~. \end{aligned}$$

In the Euclidean case, we have the following existence result.

Theorem A.2

Let $k>0$ and $K\in C^1({\mathbb {R}}^2)$ be given. Assume that $F^{K}_k$ has a stable critical point in an open set $A\Subset {\mathbb {R}}^2$. Then, for every $\varepsilon \in {\mathbb {R}}$ close enough to 0, there exists an embedded $(k+\varepsilon K)$-loop $u^\varepsilon :{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$.

Moreover, any sequence $\varepsilon _h\rightarrow 0$ has a subsequence $\varepsilon _{h_j}$ such that $u^{\varepsilon _{h_j}}\rightarrow \omega _{z_0}$ in $C^2({\mathbb {S}}^1,{\mathbb {R}}^2)$ as $j\rightarrow \infty$, where $z_0\in A$ is a critical point for $F^{K}_k$.

Sketch of the proof

We introduce the three-dimensional space of embedded solutions to the unperturbed problem, namely

$$\begin{aligned} {\mathcal {S}}=\big \{ \omega \circ \xi +z~|~\xi \in {\mathbb {S}}^1~,~~z\in {\mathbb {R}}^2~\big \}, \end{aligned}$$

and the functions $J_\varepsilon : C^2({\mathbb {R}},{\mathbb {R}}^2){\setminus }{\mathbb {R}}^2\rightarrow C^0({\mathbb {R}},{\mathbb {R}}^2)$, $\varepsilon \in {\mathbb {R}}$, given by

$$\begin{aligned} J_\varepsilon (u)=-u'' +L(u)(k+\varepsilon K(u))~ iu'= J_0(u)+L(u) K(u)~ iu'. \end{aligned}$$

We have ${\mathcal {S}}\subset \{J_0=0\}$. Since $\displaystyle {J'_0(\omega +z)\varphi = -\varphi ''+~i\varphi '-k^2\big ({\mathop {\fint }\limits _{{\mathbb {S}}^1}}\varphi \cdot \omega ~\hbox {d}x\big )\omega }$, it is quite easy to check that

$$\begin{aligned} T_{\omega +z}{\mathcal {S}}=\langle \omega ', e_1, e_2\rangle =\ker J'_0(\omega +z) \end{aligned}$$

and that $J'_0(\omega +z):T_{\omega +z}{\mathcal {S}}^\perp \rightarrow T_{\omega +z}{\mathcal {S}}^\perp$ is invertible. The remaining part of the proof runs with minor changes. $\square$

Theorem 4.3 has its Euclidean correspondence as well. We omit the proof of the next result.

Theorem A.3

Let $K\in C^1({\mathbb {R}}^2)$. Assume that K has a stable critical point in an open set $A\Subset {\mathbb {R}}^2$. Then, there exists $k_0>1$ such that for any fixed $k>k_0$, and for every $\varepsilon$ close enough to 0, there exists an embedded $(k+\varepsilon K)$-loop $u^{k,\varepsilon }:{\mathbb {S}}^1\rightarrow {\mathbb {R}}^2$.

Moreover, there exist sequences $k_h\rightarrow \infty$, $\varepsilon _h\rightarrow 0$ such that $u^{k_h,\varepsilon _{h_j}}\rightarrow \omega _{z_0}$ in $C^2({\mathbb {S}}^1,{\mathbb {R}}^2)$ as $j\rightarrow \infty$, where $z_0\in A$ is a critical point for K.