Tema 9: Busemann functions and horospherical foliations
We will always assume that $X$ is a Hadamard manifold, that is, a simply connected complete Riemannian manifold of nonpositive curvature. We consider the topology of uniform convergence on bounded subsets.
Recall again that a function $f\colon X\to\mathbb R$ is convex if for any geodesic $c\colon X\to\mathbb R$ the composition $f\circ c\colon \mathbb R\to\mathbb R$ is convex. A set $B\subset X$ is convex if for any pair $p,q\in B$ the connecting geodesic is in $B$.
Note that the pointwise limit of convex functions is convex.
Example 1. The distance function $d\colon X\times X\to\mathbb R$ is convex.
Example 2. Given $y\in X$, $x\mapsto d_y(x):= d(y,x)$ is convex and Lipschitz with Lipschitz constant 1. Indeed, $$ d(y,x)-d(y,z)\le d(x,z), \quad d(y,z)-d(y,x)\le d(x,z). $$
Example 3. If $\sigma_1,\sigma_2\colon I\to X$ are geodesics, then $t\in I\mapsto d(\sigma_1(t),\sigma_2(t))$ is convex.
Example 4. If $\sigma_{x,x'},\sigma_{y,y'}\subset X$ are geodesics rays connecting $x$ and $x'$ and $y$ and $y'$, respectively, then $d_H(\sigma_{x,x'},\sigma_{y,y'})\le \max\{d(x,y),d(x',y')\}$.
Example 5. Given $y\in X$, the function $x\mapsto d(x,y)$ is convex and 1-Lipschitz. Hence, given some geodesic $\sigma\colon\mathbb R\to X$, for every $t\ge0$, $$ x\mapsto d(\sigma(t),x)-t $$ is convex. It follows that (using the existence of the limit, shown below) the function $$ x\mapsto h(x):=\lim_{t\to\infty}d(x,\sigma(t))-t $$ is convex and 1-Lipschitz. For every $x\in X$, consider the geodesic connecting $x$ with $\sigma(t)$, its intersection points with $\partial B_r(x)$ contains exactly two points $x_1^t,x_2^t$ satisfying $d(x,x_1^t)=r=d(x,x_2^t)$ and hence $$ \Big(d(x_1^t,\sigma(t))-t\Big)-\Big(d(x_2^t,\sigma(t))-t\Big) = \Big(d(x,\sigma(t))+r-t\Big)-\Big(d(x,\sigma(t))-r-t\Big) = 2r $$ Taking accumulation points $x_i$ of $x_i^t$ as $t\to\infty$, $i=1,2$, for every $x\in X$ there are points $x_1,x_2\in\partial B_r(x)$ with $$ \lvert h(x_1)-h(x_2)\rvert =2r. $$
Recall that two geodesics $\sigma_1,\sigma_2\colon\mathbb B\to X$ are (forward) asymptotic if $$ d_H(\sigma_1(\mathbb R^+),\sigma_2(\mathbb R^+))<\infty, $$ where $d_H$ denotes the Hausdorff distance. In other words, this defines a relation between geodesic rays. Note that being forward asymptotic is an equivalence relation. The set of equivalence classes of asymptotic geodesics is called sphere at infinity or also boundary at infinity and denoted by $X(\infty)$. See the previous post (Tema 8). Given a geodesic ray $\sigma\colon\mathbb R^+\to X$, denote by $\sigma(\infty)$ the equivalence class it is contained in.
Lemma 1. For every $\xi\in X(\infty)$ and $x\in X$ there is at most one geodesic ray such that $\sigma(0)=x$ and $\sigma(\infty)=\xi$.
Proof. Recall that $t\mapsto d(\sigma_1(t),\sigma_2(t))$ is convex. Hence, $\sigma_1(0)=\sigma_2(0)$ and $\sigma_1(\infty)=\sigma_2(\infty)$ implies $\sigma_1=\sigma_2$.
We will use the following lemma.
Lemma 2. (see [Lemma 2.1, Ballmann]) Given a geodesic ray $\sigma\colon\mathbb R^+\to X$ and $x\in X$, for $n\in\mathbb N$ denote by $\sigma_n\colon[0,d(x,\sigma(n)]\to X$ the geodesic segment from $x$ to $\sigma(n)$. Then for every $R>0$ and $\varepsilon>0$, for $n,m\in\mathbb N$ sufficiently large, it holds $$ d(\sigma_n(t),\sigma_m(t))<\varepsilon \quad\text{ for all }t\in[0,R]. $$ Moreover, $\sigma_n$ converges to a geodesic ray $\sigma_{x,\xi}\colon\mathbb R^+\to X$ which is asymptotic to $\sigma$: for every $R>0$ and $\varepsilon>0$, for $n$ sufficiently large $$ d(\sigma_{x,\xi}(t),\sigma_n(t))<\varepsilon \quad\text{ for all }t\in[0,R]. $$
The following is now an immediate consequence.
Lemma 3. For every $\xi\in X(\infty)$ and $x\in X$ there is a unique geodesic ray $\sigma_{x,\xi}\colon\mathbb R^+\to X$ such that $\sigma_{x,\xi}(0)=x$ and $\sigma_{x,\xi}(\infty)=\xi$.
Note that Lemmas 1-3 correspond Lemma 1 in tema8.
Lemma 4. Let $\sigma\colon\mathbb R\to\infty$ be a geodesic, $y:=\sigma(0)$, and $\xi=\sigma(\infty)$. Then for every $x\in X$ the limit $$ b_y(x,\xi) := \lim_{t\to\infty}d(x,\sigma(t))-t $$ exists.
Proof. Check that the triangle inequality and $d(y,\sigma(t))=t$ together imply $$ 0 = d(y,\sigma(t))-t \le d(y,x)+d(x,\sigma(t))-t, $$ which implies $$ -d(y,x) \le d(x,\sigma(t))-t. $$ On the other hand, $$ d(x,\sigma(t)) \le d(x,y)+d(y,\sigma(t)) = d(x,y)+t $$ and for $t\le s$ $$\begin{split} d(x,\sigma(s))-s &\le d(x,\sigma(t))+d(\sigma(t),\sigma(s))-s = d(x,\sigma(t))+(s-t)-s \\ &= d(x,\sigma(t)) -t \le d(x,y). \end{split}$$ Hence, the sequence is nonincreasing and bounded from below, hence converging.
We showed above that $b_y(\cdot,\xi)$ is convex and Lipschitz with Lipschitz constant 1. For the following, we largely follow [Lecture I Chapter 3, Ballmann-Gromov-Schröder].
Lemma 5. For every $x\in X$ and $\xi\in X(\infty)$, the function $h=b_y(\cdot,\xi)\colon X\to\mathbb R$
- is convex,
- is Lipschitz with Lipschitz constant 1,
- for $x\in X$, $r>0$ there are unique $x_1,x_2\in \partial B_r(x)$ such that $\lvert h(x_1)-h(x_2)\rvert=2r$,
- is $C^1$ with $\lVert{\rm grad} h\rVert=1$,
- is $C^2$.
Proof.The first three properties were shown in Example 5. To prove the forth one, fix $r>0$ and consider the unit speed vector field $x\mapsto\eta(x)$ such that $\eta(x)$ is the unique unit speed vector of the unit speed geodesic from $x$ to $x_1\in\partial B_r(x)$ that $h(x_1)=h(x)+r$. As $x_1=x_1(x,r)$ is unique, this is a continuous vector field $\eta$ on $X$. Let $c\colon[-\varepsilon,\varepsilon]\to B_r(x)$ be a unit speed geodesic with $c(0)=x$. We prove that $h\circ c$ is differentiable and satisfies $(h\circ c)'(0) =\langle \dot c(0),\eta(x)\rangle$. Hence, by continuity of $\eta$, $h$ is differentiable with ${\rm grad} h= \eta$, proving the claim. To simplify notation, adding a constant, let us assume that $h(x)=0$, $h(x_1)=r$, and $h(x_2)=-r$. Lipschitz continuity of $h$ implies \[ \lvert h(c(s))-h(x_i)\rvert \le d(c(s),x_i) \] and hence \[ -d(c(s),x_1)+h(x_1) \le h(c(s)) \le d(c(s),x_2)+h(x_2), \] which implies \[ -d(c(s),x_1)+r := h_1(s) \le h(c(s)) \le h_2(s) := d(c(s),x_2)-r. \] Note that $h(c(0))=h(x)=h_1(0)=h_2(0)$. The first variation formula implies \[ h_1'(0) = \frac{d}{ds}h_1(s)|_{s=0} = \langle \dot c(0),\eta(x)\rangle = h_2'(0). \] This shows that $h\circ c$ is differentiable and $(h\circ c)'(0) =\langle \dot c(0),\eta(x)\rangle$, proving the claim. To conclude the proof, just note that $h$ is $C^2$ was shown in [Heintze-Im Hof '77] (see also [Proposition 3.2, Ballmann '95).
Given $y\in X$ and $\xi\in X(\infty)$, the function \[ x\mapsto b_y(x,\xi) \] is the Busemann function associated to $y$ and $\xi$. The level sets \[ H(\alpha) := \{x\colon b_y(x,\xi)=\alpha\} \] are the horospheres based at $\xi\in X(\infty)$. Note that \[ \sigma_{y,\xi}(\alpha)\in H(\alpha). \] The level sets only depend on $\xi$ and not on $y$. In particular, ${\rm grad} b_y(x,\xi)$ does not depend on $y$. The value $b_y(x,\xi)$ is the signed distance of $x$ from the horosphere through $y$ or, in other words, the distance between the horospheres through $x$ and $y$, respectively.
Denote by ${\rm Hor}_{x,\xi}$ the horosphere based at $\xi$ through $x\in X$. Given $v\in SX$, denote also by ${\rm Hor}_v$ the horosphere based at $c_v(\infty)\in X(\infty)$ through $c_v(0)\in X$. To take a dynamical systems-point of view, let us introduce the so-called horospherical foliations. Given $v\in SX$, let $y:= c_v(0)$, $\xi^+:= c_v(\infty)$ and $\xi^-:= c_v(-\infty)$. Then \[\begin{split} W^{cs}(v) := \{w\in SX\colon c_w(\infty)=c_v(\infty)\} = \{-{\rm grad} b_y(x,\xi^+)\colon x\in X\},\\ W^{cu}(v) := \{w\in SX\colon c_w(-\infty)=c_v(-\infty)\} = \{{\rm grad} b_y(x,\xi^-)\colon x\in X\} \end{split}\] is the weak stable or center stable leaf through $v$ and the weak unstable leaf or center unstable leaf through $v$, respectively. Each of them is subfoliated by un-/stable leafs. Here \[\begin{split} W^s(v) &:=\{-{\rm grad} b_y(x,\xi^+)\colon x\in {\rm Hor}_v\},\\ W^u(v) &:= \{{\rm grad} b_y(x,\xi^-)\colon x\in {\rm Hor}_{-v}\}, \end{split}\] define the stable and unstable leaf through $v$, respectively.