## Lower bounds on volumes of hyperbolic Haken 3-manifolds

HTML articles powered by AMS MathViewer

- by Ian Agol, Peter A. Storm and William P. Thurston; with an appendix by Nathan Dunfield PDF
- J. Amer. Math. Soc.
**20**(2007), 1053-1077 Request permission

## Abstract:

We prove a volume inequality for 3-manifolds having $C^{0}$ metrics “bent” along a surface and satisfying certain curvature conditions. The result makes use of Perelman’s work on the Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.## References

- Ian Agol,
*Volume change under drilling*, Geom. Topol.**6**(2002), 905–916. MR**1943385**, DOI 10.2140/gt.2002.6.905 - Michael Atiyah,
*Quantum field theory and low-dimensional geometry*, Progr. Theoret. Phys. Suppl.**102**(1990), 1–13 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR**1182158**, DOI 10.1143/PTPS.102.1 - Thierry Aubin,
*Some nonlinear problems in Riemannian geometry*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1636569**, DOI 10.1007/978-3-662-13006-3 - Laurent Bessières,
*Sur le volume minimal des variétés ouvertes*, Ann. Inst. Fourier (Grenoble)**50**(2000), no. 3, 965–980 (French, with English and French summaries). MR**1779901**, DOI 10.5802/aif.1780 - Gérard Besson, Gilles Courtois, and Sylvestre Gallot,
*Lemme de Schwarz réel et applications géométriques*, Acta Math.**183**(1999), no. 2, 145–169 (French). MR**1738042**, DOI 10.1007/BF02392826 - Hubert L. Bray,
*Proof of the Riemannian Penrose inequality using the positive mass theorem*, J. Differential Geom.**59**(2001), no. 2, 177–267. MR**1908823** - Martin Bridgeman,
*Bounds on volume increase under Dehn drilling operations*, Proc. London Math. Soc. (3)**77**(1998), no. 2, 415–436. MR**1635161**, DOI 10.1112/S0024611598000513 - Chun Cao and G. Robert Meyerhoff,
*The orientable cusped hyperbolic $3$-manifolds of minimum volume*, Invent. Math.**146**(2001), no. 3, 451–478. MR**1869847**, DOI 10.1007/s002220100167 - Huai-Dong Cao and Xi-Ping Zhu,
*A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow*, Asian J. Math.**10**(2006), no. 2, 165–492. MR**2233789**, DOI 10.4310/AJM.2006.v10.n2.a2 - Marc Culler and Peter B. Shalen,
*Volumes of hyperbolic Haken manifolds. I*, Invent. Math.**118**(1994), no. 2, 285–329. MR**1292114**, DOI 10.1007/BF01231535 - D. B. A. Epstein and A. Marden,
*Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces*, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR**903852** - Michael Freedman, Joel Hass, and Peter Scott,
*Least area incompressible surfaces in $3$-manifolds*, Invent. Math.**71**(1983), no. 3, 609–642. MR**695910**, DOI 10.1007/BF02095997 - Michael H. Freedman, Alexei Kitaev, Chetan Nayak, Johannes K. Slingerland, Kevin Walker, and Zhenghan Wang,
*Universal manifold pairings and positivity*, Geom. Topol.**9**(2005), 2303–2317. MR**2209373**, DOI 10.2140/gt.2005.9.2303 - David Gabai and William H. Kazez,
*Group negative curvature for $3$-manifolds with genuine laminations*, Geom. Topol.**2**(1998), 65–77. MR**1619168**, DOI 10.2140/gt.1998.2.65 - David Gabai, G. Robert Meyerhoff, and Nathaniel Thurston,
*Homotopy hyperbolic 3-manifolds are hyperbolic*, Ann. of Math. (2)**157**(2003), no. 2, 335–431. MR**1973051**, DOI 10.4007/annals.2003.157.335
Snap Oliver Goodman, - Richard S. Hamilton,
*Non-singular solutions of the Ricci flow on three-manifolds*, Comm. Anal. Geom.**7**(1999), no. 4, 695–729. MR**1714939**, DOI 10.4310/CAG.1999.v7.n4.a2 - Joel Hass,
*Acylindrical surfaces in $3$-manifolds*, Michigan Math. J.**42**(1995), no. 2, 357–365. MR**1342495**, DOI 10.1307/mmj/1029005233 - Joel Hass, J. Hyam Rubinstein, and Shicheng Wang,
*Boundary slopes of immersed surfaces in 3-manifolds*, J. Differential Geom.**52**(1999), no. 2, 303–325. MR**1758298** - Joel Hass and Peter Scott,
*The existence of least area surfaces in $3$-manifolds*, Trans. Amer. Math. Soc.**310**(1988), no. 1, 87–114. MR**965747**, DOI 10.1090/S0002-9947-1988-0965747-6 - Craig D. Hodgson and Steven P. Kerckhoff,
*Universal bounds for hyperbolic Dehn surgery*, Ann. of Math. (2)**162**(2005), no. 1, 367–421. MR**2178964**, DOI 10.4007/annals.2005.162.367 - William H. Jaco and Peter B. Shalen,
*Seifert fibered spaces in $3$-manifolds*, Mem. Amer. Math. Soc.**21**(1979), no. 220, viii+192. MR**539411**, DOI 10.1090/memo/0220 - Klaus Johannson,
*Homotopy equivalences of $3$-manifolds with boundaries*, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR**551744**, DOI 10.1007/BFb0085406 - Marc Lackenby,
*The volume of hyperbolic alternating link complements*, Proc. London Math. Soc. (3)**88**(2004), no. 1, 204–224. With an appendix by Ian Agol and Dylan Thurston. MR**2018964**, DOI 10.1112/S0024611503014291 - Bernhard Leeb,
*$3$-manifolds with(out) metrics of nonpositive curvature*, Invent. Math.**122**(1995), no. 2, 277–289. MR**1358977**, DOI 10.1007/BF01231445 - Pengzi Miao,
*Positive mass theorem on manifolds admitting corners along a hypersurface*, Adv. Theor. Math. Phys.**6**(2002), no. 6, 1163–1182 (2003). MR**1982695**, DOI 10.4310/ATMP.2002.v6.n6.a4 - Yosuke Miyamoto,
*Volumes of hyperbolic manifolds with geodesic boundary*, Topology**33**(1994), no. 4, 613–629. MR**1293303**, DOI 10.1016/0040-9383(94)90001-9 - John W. Morgan,
*On Thurston’s uniformization theorem for three-dimensional manifolds*, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 37–125. MR**758464**, DOI 10.1016/S0079-8169(08)61637-2 - Walter D. Neumann and Don Zagier,
*Volumes of hyperbolic three-manifolds*, Topology**24**(1985), no. 3, 307–332. MR**815482**, DOI 10.1016/0040-9383(85)90004-7
Per03 Grisha Perelman, - Andrew Przeworski,
*A universal upper bound on density of tube packings in hyperbolic space*, J. Differential Geom.**72**(2006), no. 1, 113–127. MR**2215457** - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419** - Richard Schoen,
*Estimates for stable minimal surfaces in three-dimensional manifolds*, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR**795231** - Richard M. Schoen,
*Variational theory for the total scalar curvature functional for Riemannian metrics and related topics*, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR**994021**, DOI 10.1007/BFb0089180 - Takashi Shioya and Takao Yamaguchi,
*Volume collapsed three-manifolds with a lower curvature bound*, Math. Ann.**333**(2005), no. 1, 131–155. MR**2169831**, DOI 10.1007/s00208-005-0667-x - Miles Simon,
*Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature*, Comm. Anal. Geom.**10**(2002), no. 5, 1033–1074. MR**1957662**, DOI 10.4310/CAG.2002.v10.n5.a7 - Teruhiko Soma,
*The Gromov invariant of links*, Invent. Math.**64**(1981), no. 3, 445–454. MR**632984**, DOI 10.1007/BF01389276 - È. B. Vinberg,
*Volumes of non-Euclidean polyhedra*, Uspekhi Mat. Nauk**48**(1993), no. 2(290), 17–46 (Russian); English transl., Russian Math. Surveys**48**(1993), no. 2, 15–45. MR**1239859**, DOI 10.1070/RM1993v048n02ABEH001011
SnapPea Jeffery Weeks,

Ag3 Ian Agol, *Lower bounds on volumes of hyperbolic Haken 3-manifolds*, arXiv:math.GT/

9906182

.

*Snap: a computer program for studying arithmetic invariants of hyperbolic 3-manifolds*, www.ms.unimelb.edu.au/~snap.

KleinerLott06 Bruce Kleiner and John Lott, *Notes on Perelman’s papers*, arXiv:math.DG/0605667

.

MorganTian06 John W. Morgan and Gang Tian, *Ricci flow and the poincaré conjecture*, 2006, arXiv:math.DG/0607607

, preprint.

*Ricci flow with surgery on three-manifolds*, arXiv:math.DG/0303109. Per02 —,

*The entropy formula for the Ricci flow and its geometric applications*, arXiv: math.DG/0211159.

Storm04 Peter A. Storm, *Hyperbolic convex cores and simplicial volume*, arXiv:math.GT/0409312

.

Th William P. Thurston,*The geometry and topology of 3-manifolds*, Lecture notes from Princeton University, 1978–80.

*SnapPea: A computer program for creating and studying hyperbolic 3-manifolds*, www.geometrygames.org.

## Additional Information

**Ian Agol**- Affiliation: Department of Mathematics, Computer Science, and Statistics, University of Illinois at Chicago, 322 SEO, m/c 249, 851 S. Morgan St., Chicago, Illinois 60607-7045
- Address at time of publication: Department of Mathematics, University of California at Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840
- MR Author ID: 671767
- ORCID: 0000-0002-4254-8483
- Email: agol@math.uic.edu, ianagol@gmail.com
**Peter A. Storm**- Affiliation: Department of Mathematics, Stanford University, Building 380, 450 Serra Mall, Stanford, California 94305-2125
- Email: storm@math.stanford.edu
**William P. Thurston**- Affiliation: Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New York 14853-4201
- Email: wpt@math.cornell.edu
**Nathan Dunfield**- Affiliation: Department of Mathematics, 253-37, Caltech, Pasadena, California 91125
- Address at time of publication: (August 1, 2007) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 341957
- ORCID: 0000-0002-9152-6598
- Email: dunfield@caltech.edu, nathan@dunfield.info
- Received by editor(s): June 30, 2005
- Published electronically: May 31, 2007
- Additional Notes: The first author was partially supported by NSF grant DMS-0204142 and the Sloan Foundation

The second author was partially supported by an NSF postdoctoral fellowship

The third author was partially supported by the NSF grant DMS-0343694

The last author was partially supported by the NSF grant DMS-0405491 and the Sloan foundation - © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**20**(2007), 1053-1077 - MSC (2000): Primary 58Jxx, 57Mxx
- DOI: https://doi.org/10.1090/S0894-0347-07-00564-4
- MathSciNet review: 2328715