Busemann function

A concept of function which measures the distance to a point at infinity. Let $M$ be a Riemannian manifold. The Riemannian metric induces a distance function $d$ on $M$. Let $\gamma$ be a ray in $M$, i.e., a unit-speed geodesic line $\gamma : [ 0 , \infty ) \rightarrow M$ such that $d ( \gamma ( t ) , \gamma ( 0 ) ) = t$ for all $t \geq 0$. The Busemann function $b _ { \gamma } : M \rightarrow \mathbf R$ with respect to $\gamma$ is defined by

\begin{equation*} b _ { \gamma } ( x ) = \operatorname { lim } _ { t \rightarrow \infty } ( t - d ( x , \gamma ( t ) ) ) , \quad x \in M. \end{equation*}

Since $t - d ( x , \gamma ( t ) )$ is bounded above by $d ( x , \gamma ( 0 ) )$ and is monotone non-decreasing in $t$, the limit always exits. It follows that $ b _ { \gamma }$ is a Lipschitz function with Lipschitz constant $1$. The level surfaces $b _ { \gamma } ^ { - 1 } ( t )$ of a Busemann function are called horospheres. Busemann functions can also be defined on intrinsic (or length) metric spaces, in the same manner. Actually, H. Busemann [a2] first introduced them on so-called $G$-spaces and used them to state the parallel axiom on straight $G$-spaces (cf. also Closed geodesic).

If $M$ has non-negative sectional curvature, $ b _ { \gamma }$ is convex, see [a3]. If $M$ has non-negative Ricci curvature, $ b _ { \gamma }$ is a subharmonic function, see [a4]. If $M$ is a Kähler manifold with non-negative holomorphic bisectional curvature, $ b _ { \gamma }$ is a plurisubharmonic function, see [a7]. If $M$ is a Hadamard manifold, $ b _ { \gamma }$ is a $C ^ { 2 }$ concave function, see [a9], [a2], and, moreover, the horospheres are $C ^ { 2 }$-hypersurfaces, see [a9]. On the Poincaré model $H ^ { 2 }$ of the hyperbolic space, the horospheres coincide with the Euclidean spheres in $H ^ { 2 }$ which are tangent to the sphere at infinity. On Hadamard manifolds, it is more customary to call $- b _ { \gamma }$ the Busemann function instead of $ b _ { \gamma }$.

More recently, M. Gromov [a1] introduced a generalization of the concept of Busemann function called the horofunction. Let $C ( N )$ be the set of continuous functions on $M$ and let $C_{ * } ( M )$ the quotient space of $C ( N )$ modulo the constant functions. Use the topology on $C ( N )$ induced from the uniform convergence on compact sets and its quotient topology on $C_{ * } ( M )$. The embedding of $M$ into $C ( N )$ defined by $M \ni x \mapsto d ( x ,\, . ) \in C ( M )$ induces an embedding $\iota : M \rightarrow C_{*} ( M )$. The closure of the image $\iota ( M )$ is a compactification of $M$ (cf. also Compactification). According to [a1], [a8], a horofunction is defined to be a class (or an element of a class) in the topological boundary $\partial \iota ( M )$ of $\iota ( M )$ in $C_{ * } ( M )$. Any Busemann function is a horofunction. For Hadamard manifolds, any horofunction can be represented as some Busemann function, see [a1]. However, this is not necessarily true for non-Hadamard manifolds. Horofunctions have been defined not only for Riemannian manifolds but also for complete locally compact metric spaces.

Let $M$ be a complete non-compact Riemannian manifold with non-negative sectional curvature, and for $p \in M$, let $b _ { p } ( x ) = \operatorname { sup } _ { \gamma } b _ { \gamma } ( x )$, $x \in M$, where $\gamma$ runs over all rays emanating from $p$. Then, $b _ { p }$ is a convex exhaustion function, see [a3], that is, a function $f$ on $M$ such that $f ^ { - 1 } ( ( - \infty , t ] )$ is compact for any $t \in f ( M )$. The function $b _ { p }$ plays an important role in the first step of the Cheeger–Gromoll structure theory for $M$, see [a3]. Any Kähler manifold admitting a strictly plurisubharmonic exhaustion function is a Stein manifold, see [a5]. This, together with the use of Busemann functions or $b _ { p }$, yields various sufficient conditions for a Kähler manifold to be Stein; see, for example, [a7], [a17]. Some results for the exhaustion property of Busemann functions are known, see [a14], [a15], [a12], [a13]. For a generalization of the notion of a horofunction and of $b _ { p }$, see [a18]. A general reference for Busemann function and its related topics is [a16].

References

[a1]	W. Ballmann, M. Gromov, V. Schroeder, "Manifolds of nonpositive curvature" , Progr. Math. , 61 , Birkhäuser (1985)
[a2]	H. Busemann, "The geometry of geodesics" , Acad. Press (1955)
[a3]	J. Cheeger, D. Gromoll, "On the structure of complete manifolds of nonnegative curvature" Ann. of Math. (2) , 96 (1972) pp. 413–443
[a4]	J. Cheeger, D. Gromoll, "The splitting theorem for manifolds of nonnegative Ricci curvature" J. Diff. Geom. , 6 (1971/72) pp. 119–128
[a5]	F. Docquier, H. Grauert, "Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten" Math. Ann. , 140 (1960) pp. 94–123
[a6]	P. Eberlein, B. O'Neill, "Visibility manifolds" Pacific J. Math. , 46 (1973) pp. 45–109
[a7]	R.E. Greene, H. Wu, "On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs" Abh. Math. Sem. Univ. Hamburg , 47 (1978) pp. 171–185 (Special issue dedicated to the seventieth birthday of Erich Käler)
[a8]	M. Gromov, "Structures métriques pour les variétés riemanniennes" , Textes Mathématiques [Mathematical Texts] , 1 , CEDIC (1981) (Edited by J. Lafontaine and P. Pansu)
[a9]	E. Heintze, H.-C. Im Hof, "Geometry of horospheres" J. Diff. Geom. , 12 : 4 (1977) pp. 481–491 (1978)
[a10]	N. Innami, "On the terminal points of co-rays and rays" Arch. Math. (Basel) , 45 : 5 (1985) pp. 468–470
[a11]	N. Innami, "Differentiability of Busemann functions and total excess" Math. Z. , 180 : 2 (1982) pp. 235–247
[a12]	A. Kasue, "A compactification of a manifold with asymptotically nonnegative curvature" Ann. Sci. Ecole Norm. Sup. 4 , 21 : 4 (1988) pp. 593–622
[a13]	Z. Shen, "On complete manifolds of nonnegative $k$th-Ricci curvature" Trans. Amer. Math. Soc. , 338 : 1 (1993) pp. 289–310
[a14]	K. Shiohama, "Busemann functions and total curvature" Invent. Math. , 53 : 3 (1979) pp. 281–297
[a15]	K. Shiohama, "The role of total curvature on complete noncompact Riemannian $2$-manifolds" Illinois J. Math. , 28 : 4 (1984) pp. 597–620
[a16]	K. Shiohama, "Topology of complete noncompact manifolds" , Geometry of Geodesics and Related Topics (Tokyo, 1982) , Adv. Stud. Pure Math. , 3 , North-Holland (1984) pp. 423–450
[a17]	Y.T. Siu, S.T. Yau, "Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay" Ann. of Math. (2) , 105 : 2 (1977) pp. 225–264
[a18]	H. Wu, "An elementary method in the study of nonnegative curvature" Acta Math. , 142 : 1–2 (1979) pp. 57–78