View source for Busemann function

A concept of function which measures the distance to a point at infinity. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205501.png" /> be a [[Riemannian manifold|Riemannian manifold]]. The Riemannian metric induces a distance function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205502.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205503.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205504.png" /> be a ray in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205505.png" />, i.e., a unit-speed [[Geodesic line|geodesic line]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205506.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205507.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205508.png" />. The Busemann function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b1205509.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055010.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055011.png" /></td> </tr></table>

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055012.png" /> is bounded above by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055013.png" /> and is monotone non-decreasing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055014.png" />, the limit always exits. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055015.png" /> is a Lipschitz function with [[Lipschitz constant|Lipschitz constant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055016.png" />. The level surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055017.png" /> 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 [[#References|[a2]]] first introduced them on so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055020.png" />-spaces and used them to state the parallel axiom on straight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055021.png" />-spaces (cf. also [[Closed geodesic|Closed geodesic]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055022.png" /> has non-negative [[Sectional curvature|sectional curvature]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055023.png" /> is convex, see [[#References|[a3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055024.png" /> has non-negative [[Ricci curvature|Ricci curvature]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055025.png" /> is a [[Subharmonic function|subharmonic function]], see [[#References|[a4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055026.png" /> is a [[Kähler manifold|Kähler manifold]] with non-negative holomorphic bisectional curvature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055027.png" /> is a [[Plurisubharmonic function|plurisubharmonic function]], see [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055028.png" /> is a Hadamard manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055029.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055030.png" /> concave function, see [[#References|[a9]]], [[#References|[a2]]], and, moreover, the horospheres are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055031.png" />-hypersurfaces, see [[#References|[a9]]]. On the [[Poincaré model|Poincaré model]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055032.png" /> of the hyperbolic space, the horospheres coincide with the Euclidean spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055033.png" /> which are tangent to the sphere at infinity. On Hadamard manifolds, it is more customary to call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055034.png" /> the Busemann function instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055035.png" />.

More recently, M. Gromov [[#References|[a1]]] introduced a generalization of the concept of Busemann function called the horofunction. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055036.png" /> be the set of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055037.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055038.png" /> the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055039.png" /> modulo the constant functions. Use the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055040.png" /> induced from the [[Uniform convergence|uniform convergence]] on compact sets and its quotient topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055041.png" />. The embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055042.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055043.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055044.png" /> induces an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055045.png" />. The closure of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055046.png" /> is a compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055047.png" /> (cf. also [[Compactification|Compactification]]). According to [[#References|[a1]]], [[#References|[a8]]], a horofunction is defined to be a class (or an element of a class) in the topological boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055050.png" />. Any Busemann function is a horofunction. For Hadamard manifolds, any horofunction can be represented as some Busemann function, see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055051.png" /> be a complete non-compact Riemannian manifold with non-negative sectional curvature, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055052.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055055.png" /> runs over all rays emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055056.png" />. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055057.png" /> is a convex exhaustion function, see [[#References|[a3]]], that is, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055060.png" /> is compact for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055061.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055062.png" /> plays an important role in the first step of the Cheeger–Gromoll structure theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055063.png" />, see [[#References|[a3]]]. Any Kähler manifold admitting a strictly plurisubharmonic exhaustion function is a [[Stein manifold|Stein manifold]], see [[#References|[a5]]]. This, together with the use of Busemann functions or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055064.png" />, yields various sufficient conditions for a Kähler manifold to be Stein; see, for example, [[#References|[a7]]], [[#References|[a17]]]. Some results for the exhaustion property of Busemann functions are known, see [[#References|[a14]]], [[#References|[a15]]], [[#References|[a12]]], [[#References|[a13]]]. For a generalization of the notion of a horofunction and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055065.png" />, see [[#References|[a18]]]. A general reference for Busemann function and its related topics is [[#References|[a16]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Ballmann,   M. Gromov,   V. Schroeder,   "Manifolds of nonpositive curvature" , ''Progr. Math.'' , '''61''' , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Busemann,   "The geometry of geodesics" , Acad. Press  (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cheeger,   D. Gromoll,   "On the structure of complete manifolds of nonnegative curvature"  ''Ann. of Math. (2)'' , '''96'''  (1972)  pp. 413–443</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Cheeger,   D. Gromoll,   "The splitting theorem for manifolds of nonnegative Ricci curvature"  ''J. Diff. Geom.'' , '''6'''  (1971/72)  pp. 119–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Docquier,   H. Grauert,   "Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten"  ''Math. Ann.'' , '''140'''  (1960)  pp. 94–123</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Eberlein,   B. O'Neill,   "Visibility manifolds"  ''Pacific J. Math.'' , '''46'''  (1973)  pp. 45–109</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Gromov,   "Structures métriques pour les variétés riemanniennes" , ''Textes Mathématiques [Mathematical Texts]'' , '''1''' , CEDIC  (1981)  (Edited by J. Lafontaine and P. Pansu)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Heintze,   H.-C. Im Hof,   "Geometry of horospheres"  ''J. Diff. Geom.'' , '''12''' :  4  (1977)  pp. 481–491 (1978)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  N. Innami,   "On the terminal points of co-rays and rays"  ''Arch. Math. (Basel)'' , '''45''' :  5  (1985)  pp. 468–470</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  N. Innami,   "Differentiability of Busemann functions and total excess"  ''Math. Z.'' , '''180''' :  2  (1982)  pp. 235–247</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Kasue,   "A compactification of a manifold with asymptotically nonnegative curvature"  ''Ann. Sci. Ecole Norm. Sup. 4'' , '''21''' :  4  (1988)  pp. 593–622</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  Z. Shen,   "On complete manifolds of nonnegative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055066.png" />th-Ricci curvature"  ''Trans. Amer. Math. Soc.'' , '''338''' :  1  (1993)  pp. 289–310</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  K. Shiohama,   "Busemann functions and total curvature"  ''Invent. Math.'' , '''53''' :  3  (1979)  pp. 281–297</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  K. Shiohama,   "The role of total curvature on complete noncompact Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120550/b12055067.png" />-manifolds"  ''Illinois J. Math.'' , '''28''' :  4  (1984)  pp. 597–620</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H. Wu,   "An elementary method in the study of nonnegative curvature"  ''Acta Math.'' , '''142''' :  1–2  (1979)  pp. 57–78</TD></TR></table>