**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 [[ebsd2021: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.