Einstein solvmanifolds are standard

Abstract

We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has shown that under a simple algebraic condition (he calls such a solvmanifold standard), Einstein solvmanifolds have many remarkable structural and uniqueness properties. In this paper, we prove that any Einstein solvmanifold is standard, by applying a stratification procedure adapted from one in geometric invariant theory due to F. Kirwan.

  • [AznWls] Go to document R. Azencott and E. N. Wilson, "Homogeneous manifolds with negative curvature. I," Trans. Amer. Math. Soc., vol. 215, pp. 323-362, 1976.
    @article {AznWls, MRKEY = {0394507},
      AUTHOR = {Azencott, Robert and Wilson, Edward N.},
      TITLE = {Homogeneous manifolds with negative curvature. {I}},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {215},
      YEAR = {1976},
      PAGES = {323--362},
      ISSN = {0002-9947},
      MRCLASS = {53C30 (22E25 17B30)},
      MRNUMBER = {52 \#15308},
      MRREVIEWER = {Rolf Sulanke},
      DOI = {10.2307/1999731},
      ZBLNUMBER = {0293.53017},
      }
  • [Brg2] M. Berger, A Panoramic View of Riemannian Geometry, New York: Springer-Verlag, 2003.
    @book {Brg2, MRKEY = {2002701},
      AUTHOR = {Berger, Marcel},
      TITLE = {A {P}anoramic View of {R}iemannian {G}eometry},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      PAGES = {xxiv+824},
      ISBN = {3-540-65317-1},
      MRCLASS = {53-02 (53C20 53C21 53C55 58J20)},
      MRNUMBER = {2004h:53001},
      MRREVIEWER = {J{ü}rgen Eichhorn},
      ZBLNUMBER = {1038.53002},
      }
  • [Bss] A. L. Besse, Einstein Manifolds, New York: Springer-Verlag, 1987, vol. 10.
    @book {Bss, MRKEY = {867684},
      AUTHOR = {Besse, Arthur L.},
      TITLE = {Einstein Manifolds},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {10},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {xii+510},
      ISBN = {3-540-15279-2},
      MRCLASS = {53C25 (53-02 53C21 53C30 53C55 58D17 58E11)},
      MRNUMBER = {88f:53087},
      MRREVIEWER = {S. M. Salamon},
      ZBLNUMBER = {0613.53001},
      }
  • [BhmWngZll] Go to document C. Böhm, M. Wang, and W. Ziller, "A variational approach for compact homogene-\linebreak ous Einstein manifolds," Geom. Funct. Anal., vol. 14, iss. 4, pp. 681-733, 2004.
    @article {BhmWngZll, MRKEY = {2084976},
      AUTHOR = {B{ö}hm, C. and Wang, M. and Ziller, W.},
      TITLE = {A variational approach for compact homogene-\linebreak ous {E}instein manifolds},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {14},
      YEAR = {2004},
      NUMBER = {4},
      PAGES = {681--733},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {53C25 (53C30)},
      MRNUMBER = {2005g:53074},
      MRREVIEWER = {Megan M. Kerr},
      DOI = {10.1007/s00039-004-0471-x},
      ZBLNUMBER = {1068.53029},
      }
  • [Dtt] Go to document I. Dotti Miatello, "Ricci curvature of left invariant metrics on solvable unimodular Lie groups," Math. Z., vol. 180, iss. 2, pp. 257-263, 1982.
    @article {Dtt, MRKEY = {661702},
      AUTHOR = {Dotti Miatello, Isabel},
      TITLE = {Ricci curvature of left invariant metrics on solvable unimodular {L}ie groups},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {180},
      YEAR = {1982},
      NUMBER = {2},
      PAGES = {257--263},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {53C20 (22E25)},
      MRNUMBER = {84a:53044},
      MRREVIEWER = {Hillel Gauchman},
      DOI = {10.1007/BF01318909},
      ZBLNUMBER = {0471.53033},
      }
  • [Frd] H. Freudenthal and H. de Vries, Linear Lie Groups, New York: Academic Press, 1969, vol. 35.
    @book {Frd, MRKEY = {0260926},
      AUTHOR = {Freudenthal, Hans and de Vries, H.},
      TITLE = {Linear {L}ie Groups},
      SERIES = {Pure Appl. Math.},
      VOLUME={35},
      PUBLISHER = {Academic Press},
      ADDRESS = {New York},
      YEAR = {1969},
      PAGES = {xxii+547},
      MRCLASS = {22.50},
      MRNUMBER = {41 \#5546},
      MRREVIEWER = {W. Klingenberg},
      ZBLNUMBER = {0377.22001},
      }
  • [Hbr] Go to document J. Heber, "Noncompact homogeneous Einstein spaces," Invent. Math., vol. 133, iss. 2, pp. 279-352, 1998.
    @article {Hbr, MRKEY = {1632782},
      AUTHOR = {Heber, Jens},
      TITLE = {Noncompact homogeneous {E}instein spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {133},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {279--352},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {53C30 (53C25 53C35)},
      MRNUMBER = {99d:53046},
      MRREVIEWER = {McKenzie Y. Wang},
      DOI = {10.1007/s002220050247},
      ZBLNUMBER = {0906.53032},
      }
  • [Hss] Go to document W. H. Hesselink, "Uniform instability in reductive groups," J. Reine Angew. Math., vol. 303/304, pp. 74-96, 1978.
    @article {Hss, MRKEY = {514673},
      AUTHOR = {Hesselink, Wim H.},
      TITLE = {Uniform instability in reductive groups},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {303/304},
      YEAR = {1978},
      PAGES = {74--96},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {14L30},
      MRNUMBER = {81a:14022},
      MRREVIEWER = {Amassa Fauntleroy},
      DOI = {10.1515/crll.1978.303-304.74},
      ZBLNUMBER = {0386.20020},
      }
  • [Kmp] Go to document G. R. Kempf, "Instability in invariant theory," Ann. of Math., vol. 108, iss. 2, pp. 299-316, 1978.
    @article {Kmp, MRKEY = {506989},
      AUTHOR = {Kempf, George R.},
      TITLE = {Instability in invariant theory},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {108},
      YEAR = {1978},
      NUMBER = {2},
      PAGES = {299--316},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20G05 (14L30 14M99 15A72)},
      MRNUMBER = {80c:20057},
      MRREVIEWER = {V. L. Popov},
      DOI = {10.2307/1971168},
      ZBLNUMBER = {0406.14031},
      }
  • [Krw1] F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton, NJ: Princeton Univ. Press, 1984, vol. 31.
    @book {Krw1, MRKEY = {766741},
      AUTHOR = {Kirwan, Frances Clare},
      TITLE = {Cohomology of Quotients in Symplectic and Algebraic Geometry},
      SERIES = {Mathematical Notes},
      VOLUME = {31},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1984},
      PAGES = {i+211},
      ISBN = {0-691-08370-3},
      MRCLASS = {58F05 (14D25 14L30)},
      MRNUMBER = {86i:58050},
      MRREVIEWER = {Martin A. Guest},
      ZBLNUMBER = {0553.14020},
      }
  • [strata] Go to document J. Lauret, "On the moment map for the variety of Lie algebras," J. Funct. Anal., vol. 202, iss. 2, pp. 392-423, 2003.
    @article {strata, MRKEY = {1990531},
      AUTHOR = {Lauret, Jorge},
      TITLE = {On the moment map for the variety of {L}ie algebras},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {202},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {392--423},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {14L30 (17B30 53D20)},
      MRNUMBER = {2004d:14067},
      MRREVIEWER = {Rupert Wei Tze Yu},
      DOI = {10.1016/S0022-1236(02)00108-8},
      ZBLNUMBER = {1036.17010},
      }
  • [minimal] Go to document J. Lauret, "A canonical compatible metric for geometric structures on nilmanifolds," Ann. Global Anal. Geom., vol. 30, iss. 2, pp. 107-138, 2006.
    @article {minimal, MRKEY = {2234091},
      AUTHOR = {Lauret, Jorge},
      TITLE = {A canonical compatible metric for geometric structures on nilmanifolds},
      JOURNAL = {Ann. Global Anal. Geom.},
      FJOURNAL = {Annals of Global Analysis and Geometry},
      VOLUME = {30},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {107--138},
      ISSN = {0232-704X},
      CODEN = {AGAGDV},
      MRCLASS = {53D20 (22E25 53D05)},
      MRNUMBER = {2007g:53092},
      MRREVIEWER = {Anna M. Fino},
      DOI = {10.1007/s10455-006-9015-y},
      ZBLNUMBER = {1102.53021},
      }
  • [cruzchica] J. Lauret, "Einstein solvmanifolds and nilsolitons," in New Developments in Lie Theory and Geometry, Providence, RI: Amer. Math. Soc., 2009, vol. 491, pp. 1-35.
    @incollection {cruzchica, MRKEY = {2537049},
      AUTHOR = {Lauret, Jorge},
      TITLE = {Einstein solvmanifolds and nilsolitons},
      BOOKTITLE = {New Developments in {L}ie Theory and Geometry},
      SERIES = {Contemp. Math.},
      VOLUME = {491},
      PAGES = {1--35},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {53C25 (14L24 22E25 53C30)},
      MRNUMBER = {2010g:53081},
      MRREVIEWER = {McKenzie Y. Wang},
      ZBLNUMBER = {1186.53058},
      }
  • [einsteinsolv] J. Lauret and C. E. Will, "Einstein solvmanifolds: existence and non-existence questions," Math. Ann..
    @article {einsteinsolv, MRKEY = {2537049},
      AUTHOR = {Lauret, Jorge and Will, C. E.},
      TITLE = {Einstein solvmanifolds: existence and non-existence questions},
      JOURNAL={Math. Ann.},
      NOTE={in press},
      }
  • [LbrWng] Surveys in Differential Geometry: Essays on Einstein Manifolds, LeBrun, C. and Wang, M., Eds., Somerville, MA: International Press, 1999, vol. VI.
    @book {LbrWng, MRKEY = {1798603},
      EDITOR={Claude LeBrun and McKenzie Wang},
      TITLE = {Surveys in Differential Geometry: Essays on {E}instein Manifolds},
      SERIES = {Surveys in Differential Geometry},
      VOLUME={VI},
      NOTE = {Lectures on geometry and topology, sponsored by Lehigh University's {\it Journal of Differential Geometry}},
      PUBLISHER = {International Press},
      ADDRESS = {Somerville, MA},
      YEAR = {1999},
      PAGES = {x+423},
      ISBN = {1-57146-068-3},
      MRCLASS = {53-06 (53C25)},
      MRNUMBER = {2001f:53003},
      ZBLNUMBER={0961.00021},
      }
  • [Mmf] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., New York: Springer-Verlag, 1994, vol. 34.
    @book {Mmf, MRKEY = {1304906},
      AUTHOR = {Mumford, D. and Fogarty, J. and Kirwan, F.},
      TITLE = {Geometric Invariant Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {34},
      EDITION = {Third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {xiv+292},
      ISBN = {3-540-56963-4},
      MRCLASS = {14D25 (58E05 58F05)},
      MRNUMBER = {95m:14012},
      MRREVIEWER = {Yi Hu},
      ZBLNUMBER = {0797.14004},
      }
  • [Nss] Go to document L. Ness, "A stratification of the null cone via the moment map," Amer. J. Math., vol. 106, iss. 6, pp. 1281-1329, 1984.
    @article {Nss, MRKEY = {765581},
      AUTHOR = {Ness, Linda},
      TITLE = {A stratification of the null cone via the moment map},
      NOTE = {With an appendix by David Mumford},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {106},
      YEAR = {1984},
      NUMBER = {6},
      PAGES = {1281--1329},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {14D25 (15A72 57S25 58F12 58F18)},
      MRNUMBER = {86c:14010},
      MRREVIEWER = {Peter J. Braam},
      DOI = {10.2307/2374395},
      ZBLNUMBER = {0604.14006},
      }
  • [NktNkn] E. V. Nikitenko and Y. G. Nikonorov, "Six-dimensional Einstein solvmanifolds [Translation of Mat. Tr. 8 (2005), no. 1, 71–121; MR1955023]," Siberian Adv. Math., vol. 16, iss. 1, pp. 66-112, 2006.
    @article {NktNkn, MRKEY = {2206760},
      AUTHOR = {Nikitenko, E. V. and Nikonorov, Yu. G.},
      TITLE = {Six-dimensional {E}instein solvmanifolds [{T}ranslation of {M}at. {T}r. {\bf 8} (2005), no. 1, 71--121; MR1955023]},
      JOURNAL = {Siberian Adv. Math.},
      FJOURNAL = {Siberian Advances in Mathematics},
      VOLUME = {16},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {66--112},
      ISSN = {1055-1344},
      MRCLASS = {53C25 (53C30)},
      MRNUMBER = {2206760},
      }
  • [Nkl] Y. Nikolayevsky, Nilradicals of Einstein solvmanifolds, 2006.
    @misc{Nkl,
      author={Nikolayevsky, Y.},
      TITLE={Nilradicals of Einstein solvmanifolds},
      YEAR={2006},
      NOTE={preprint},
      ARXIV={math.DG/0612117},
      }
  • [Nkl3] Y. Nikolayevsky, "Einstein solvmanifolds and the pre-Einstein derivation," Trans. Amer. Math. Soc..
    @article{Nkl3,
      author={Nikolayevsky, Y.},
      TITLE={Einstein solvmanifolds and the pre-Einstein derivation},
      JOURNAL={Trans. Amer. Math. Soc.},
      NOTE={to appear},
      }
  • [RchSld] Go to document R. W. Richardson and P. J. Slodowy, "Minimum vectors for real reductive algebraic groups," J. London Math. Soc., vol. 42, iss. 3, pp. 409-429, 1990.
    @article {RchSld, MRKEY = {1087217},
      AUTHOR = {Richardson, R. W. and Slodowy, P. J.},
      TITLE = {Minimum vectors for real reductive algebraic groups},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {42},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {409--429},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {14L30 (32M05)},
      MRNUMBER = {92a:14055},
      MRREVIEWER = {Dennis Snow},
      DOI = {10.1112/jlms/s2-42.3.409},
      ZBLNUMBER = {0675.14020},
      }
  • [Sch] Go to document D. Schueth, "On the `standard’ condition for noncompact homogeneous Einstein spaces," Geom. Dedicata, vol. 105, pp. 77-83, 2004.
    @article {Sch, MRKEY = {2057245},
      AUTHOR = {Schueth, Dorothee},
      TITLE = {On the `standard' condition for noncompact homogeneous {E}instein spaces},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {105},
      YEAR = {2004},
      PAGES = {77--83},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {53C25 (53C30)},
      MRNUMBER = {2005e:53066},
      MRREVIEWER = {McKenzie Y. Wang},
      DOI = {10.1023/B:GEOM.0000024659.53384.3c},
      ZBLNUMBER={1059.53039},
      }

Authors

Jorge Lauret

FaMAF and CIEM
Universidad Nacional de Córdoba
Córdoba
Argentina