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]
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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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},
}
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