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--- ebsd2021:tema9 [2021/09/22 09:57] – 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 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 $\sigma(\infty)=\xi$.
@@ Line 66: / Line 68: @@
 **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
@@ Line 104: / Line 108: @@
 **Lemma 5.** For every $x\in X$ and $\xi\in X(\infty)$, the function $h=b_y(\cdot,\xi)\colon X\to\mathbb R$
-  * Unordered List Item is convex,
+  * is convex,
-  * Unordered List Itemis Lipschitz with Lipschitz constant 1,
+  * is Lipschitz with Lipschitz constant 1,
-  * Unordered List Item for $x\in X$, $r>0$ there are unique $x_1,x_2\in \partial B_r(x)$ such that $\lvert h(x_1)-h(x_2)\rvert=2r$,
+  * for $x\in X$, $r>0$ there are unique $x_1,x_2\in \partial B_r(x)$ such that $\lvert h(x_1)-h(x_2)\rvert=2r$,
-  * Unordered List Item is $C^1$ with $\lVert\grad  h\rVert=1$,
+  * is $C^1$ with $\lVert{\rm grad}  h\rVert=1$,
-  * Unordered List Item is $C^2$.
+  * 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 $\grad h= \eta$, 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
 \[
 	\lvert h(c(s))-h(x_i)\rvert
@@ Line 154: / Line 158: @@
 	\sigma_{y,\xi}(\alpha)\in H(\alpha).
 \]
-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 161: / 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 170: / 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.