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--- ebsd2021:tema9 [2021/09/22 09:56] – escola
+++ ebsd2021:tema9 [2021/09/30 12:19] (current) – escola
@@ Line 1: / Line 1: @@
+**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.
@@ Line 5: / Line 7: @@
 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 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,
+**Example 2.** Given $y\in X$, $x\mapsto d_y(x):= d(y,x)$ is convex and Lipschitz with Lipschitz constant 1. Indeed,
 $$
 	d(y,x)-d(y,z)\le d(x,z), \quad
@@ Line 13: / Line 15: @@
 $$
-**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 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 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 $t\ge0$,
+**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 $t\ge0$,
 $$
 	x\mapsto d(\sigma(t),x)-t
@@ Line 42: / Line 44: @@
 $$
 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 $X(\infty)$. Given a geodesic ray $\sigma\colon\mathbb R^+\to X$, denote by $\sigma(\infty)$ the equivalence class it is contained in.
+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 $\sigma(\infty)=\xi$.
+**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 $\sigma_1=\sigma_2$.
@@ Line 51: / Line 53: @@
 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
+**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
 $$
 	d(\sigma_n(t),\sigma_m(t))<\varepsilon
@@ Line 67: / Line 69: @@
 **Lemma 3.** For every $\xi\in X(\infty)$ and $x\in X$ there is a unique geodesic ray $\sigma_{x,\xi}\colon\mathbb R^+\to X$ such that $\sigma_{x,\xi}(0)=x$ and $\sigma_{x,\xi}(\infty)=\xi$.
-**Lemma 4** Let $\sigma\colon\mathbb R\to\infty$ be a geodesic, $y:=\sigma(0)$, and $\xi=\sigma(\infty)$. Then for every $x\in X$ the limit
+Note that Lemmas 1-3 correspond Lemma 1 in [[ebsd2021:tema8|]].
+**Lemma 4.** Let $\sigma\colon\mathbb R\to\infty$ be a geodesic, $y:=\sigma(0)$, and $\xi=\sigma(\infty)$. Then for every $x\in X$ the limit
 $$
 	b_y(x,\xi)
@@ Line 104: / Line 108: @@
 **Lemma 5.** For every $x\in X$ and $\xi\in X(\infty)$, the function $h=b_y(\cdot,\xi)\colon X\to\mathbb R$
-\begin{itemize}
+  * is convex,
-\item is convex,
+  * is Lipschitz with Lipschitz constant 1,
-\item 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 $\lvert h(x_1)-h(x_2)\rvert=2r$,
-\item 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 $\lVert{\rm grad}  h\rVert=1$,
-\item is $C^1$ with $\lVert\grad  h\rVert=1$,
+  * is $C^2$.
-\item  is $C^2$.
-\end{itemize}
-**Proof.**The first three properties were shown in Example 4. 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 $h(x_1)=h(x)+r$.
+**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 $\grad h= \eta$, 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
+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
 \[
 	\lvert h(c(s))-h(x_i)\rvert
@@ Line 155: / Line 158: @@
 	\sigma_{y,\xi}(\alpha)\in H(\alpha).
 \]
-The level sets only depend on $\xi$ and not on $y$. In particular, $\grad b_y(x,\xi)$ 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.
+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 $c_v(0)\in X$.
@@ Line 162: / Line 165: @@
 	W^{cs}(v)
 	:= \{w\in SX\colon c_w(\infty)=c_v(\infty)\}
-	= \{-\grad b_y(x,\xi^+)\colon x\in X\},\\
+	= \{-{\rm grad} b_y(x,\xi^+)\colon x\in X\},\\
 	W^{cu}(v)
 	:= \{w\in SX\colon c_w(-\infty)=c_v(-\infty)\}
-	= \{\grad b_y(x,\xi^-)\colon x\in X\}
+	= \{{\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.
@@ Line 171: / Line 174: @@
 \[\begin{split}
 	W^s(v)
-	&:=\{-\grad b_y(x,\xi^+)\colon x\in {\rm Hor}_v\},\\
+	&:=\{-{\rm grad} b_y(x,\xi^+)\colon x\in {\rm Hor}_v\},\\
 	W^u(v)
-	&:= \{\grad b_y(x,\xi^-)\colon x\in {\rm Hor}_{-v}\},
+	&:= \{{\rm grad} b_y(x,\xi^-)\colon x\in {\rm Hor}_{-v}\},
 \end{split}\]
 define the //stable// and //unstable leaf// through $v$, respectively.