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ebsd2021:tema9

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ebsd2021:tema9 [2021/09/22 09:58] – escolaebsd2021:tema9 [2021/09/30 12:19] (current) – escola
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 +**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.  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. 
  
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 $$ $$
 where $d_H$ denotes the Hausdorff distance. In other words, this defines a relation between geodesic rays.  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 $\sigma(\infty)=\xi$.
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 **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 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 **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
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 **Lemma 5.** For every $x\in X$ and $\xi\in X(\infty)$, the function $h=b_y(\cdot,\xi)\colon X\to\mathbb R$ **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 $\lVert{\rm grad}  h\rVert=1$, +  * is $C^1$ with $\lVert{\rm grad}  h\rVert=1$, 
-  * Unordered List Item is $C^2$.+  * is $C^2$.
  
  
ebsd2021/tema9.1632315530.txt.gz · Last modified: 2021/09/22 09:58 by escola