Abstract
We prove that constant scalar curvature Kähler metric “adjacent” to a fixed Kähler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the Calabi flow, from a new geometric perspective, and on the geometry of the space of Kähler metrics.
-
[D1] S. K. Donaldson, "Remarks on gauge theory, complex geometry and $4$-manifold topology," in Fields Medallists’ Lectures, River Edge, NJ: World Sci. Publ., 1997, vol. 5, pp. 384-403.
@incollection{D1, address = {River Edge, NJ},
author = {Donaldson, Simon K.},
booktitle = {Fields {M}edallists' Lectures},
pages = {384--403},
publisher = {World Sci. Publ.},
series = {World Sci. Ser. 20th Century Math.},
title = {Remarks on gauge theory, complex geometry and {$4$}-manifold topology},
volume = {5},
year = {1997},
} -
[Fu] A. Fujiki, "Moduli space of polarized algebraic manifolds and Kähler metrics [translation of Sûgaku 42 (1990), no. 3, 231–243; \mr1073369]," Sugaku Expositions, vol. 5, iss. 2, pp. 173-191, 1992.
@article{Fu,
author = {Fujiki, Akira},
journal = {Sugaku Expositions},
number = {2},
pages = {173--191},
title = {Moduli space of polarized algebraic manifolds and {K}ähler metrics [translation of {S}ûgaku {\bf 42} (1990), no. 3, 231--243; \mr{1073369}]},
volume = {5},
year = {1992},
issn = {0898-9583},
} -
[Th]
R. P. Thomas, "Notes on GIT and symplectic reduction for bundles and varieties," in Surveys in Differential Geometry. Vol. X, Somerville, MA: Int. Press, 2006, vol. 10, pp. 221-273.@incollection{Th, address = {Somerville, MA},
author = {Thomas, R. P.},
booktitle = {Surveys in Differential Geometry. {V}ol. {X}},
pages = {221--273},
publisher = {Int. Press},
series = {Surv. Differ. Geom.},
title = {Notes on {GIT} and symplectic reduction for bundles and varieties},
volume = {10},
year = {2006},
doi = {10.4310/SDG.2005.v10.n1.a7},
} -
[Ti1] G. Tian, "Kähler-Einstein metrics with positive scalar curvature," Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997.
@article{Ti1,
author = {Tian, Gang},
journal = {Invent. Math.},
number = {1},
pages = {1--37},
title = {Kähler-{E}instein metrics with positive scalar curvature},
volume = {130},
year = {1997},
doi = {10.1007/s002220050176},
issn = {0020-9910},
} -
[D5] S. K. Donaldson, "Conjectures in Kähler geometry," in Strings and Geometry, Providence, RI: Amer. Math. Soc., 2004, vol. 3, pp. 71-78.
@incollection{D5, address = {Providence, RI},
author = {Donaldson, Simon K.},
booktitle = {Strings and Geometry},
pages = {71--78},
publisher = {Amer. Math. Soc.},
series = {Clay Math. Proc.},
title = {Conjectures in {K}ähler geometry},
volume = {3},
year = {2004},
} -
[RT] J. Ross and R. Thomas, "A study of the Hilbert-Mumford criterion for the stability of projective varieties," J. Algebraic Geom., vol. 16, iss. 2, pp. 201-255, 2007.
@article{RT,
author = {Ross, Julius and Thomas, Richard},
journal = {J. Algebraic Geom.},
number = {2},
pages = {201--255},
title = {A study of the {H}ilbert-{M}umford criterion for the stability of projective varieties},
volume = {16},
year = {2007},
doi = {10.1090/S1056-3911-06-00461-9},
issn = {1056-3911},
} -
[Pa] S. T. Paul, "Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics," Ann. of Math., vol. 175, iss. 1, pp. 255-296, 2012.
@article{Pa,
author = {Paul, Sean Timothy},
journal = {Ann. of Math.},
number = {1},
pages = {255--296},
title = {Hyperdiscriminant polytopes, {C}how polytopes, and {M}abuchi energy asymptotics},
volume = {175},
year = {2012},
doi = {10.4007/annals.2012.175.1.7},
issn = {0003-486X},
} -
[D6] S. K. Donaldson, "Constant scalar curvature metrics on toric surfaces," Geom. Funct. Anal., vol. 19, iss. 1, pp. 83-136, 2009.
@article{D6,
author = {Donaldson, Simon K.},
journal = {Geom. Funct. Anal.},
number = {1},
pages = {83--136},
title = {Constant scalar curvature metrics on toric surfaces},
volume = {19},
year = {2009},
doi = {10.1007/s00039-009-0714-y},
issn = {1016-443X},
} -
[D3] S. K. Donaldson, "Scalar curvature and projective embeddings. I," J. Differential Geom., vol. 59, iss. 3, pp. 479-522, 2001.
@article{D3,
author = {Donaldson, Simon K.},
journal = {J. Differential Geom.},
number = {3},
pages = {479--522},
title = {Scalar curvature and projective embeddings. {I}},
volume = {59},
year = {2001},
issn = {0022-040X},
url = {http://projecteuclid.org/euclid.jdg/1090349449},
} -
[CT] X. Chen and G. Tian, "Geometry of Kähler metrics and foliations by holomorphic discs," Publ. Math. IHES, vol. 107, pp. 1-107, 2008.
@article{CT,
author = {Chen, X. and Tian, G.},
journal = {Publ. Math. IHES},
pages = {1--107},
title = {Geometry of {K}ähler metrics and foliations by holomorphic discs},
volume = {107},
year = {2008},
doi = {10.1007/s10240-008-0013-4},
} -
[BM] S. Bando and T. Mabuchi, "Uniqueness of Einstein Kähler metrics modulo connected group actions," in Algebraic Geometry, Sendai, 1985, Amsterdam: North-Holland, 1987, vol. 10, pp. 11-40.
@incollection{BM, address = {Amsterdam},
author = {Bando, Shigetoshi and Mabuchi, Toshiki},
booktitle = {Algebraic Geometry, {S}endai, 1985},
pages = {11--40},
publisher = {North-Holland},
series = {Adv. Stud. Pure Math.},
title = {Uniqueness of {E}instein {K}ähler metrics modulo connected group actions},
volume = {10},
year = {1987},
} -
[TZ1] G. Tian and X. Zhu, "A new holomorphic invariant and uniqueness of Kähler-Ricci solitons," Comment. Math. Helv., vol. 77, iss. 2, pp. 297-325, 2002.
@article{TZ1,
author = {Tian, Gang and Zhu, Xiaohua},
journal = {Comment. Math. Helv.},
number = {2},
pages = {297--325},
title = {A new holomorphic invariant and uniqueness of {K}ähler-{R}icci solitons},
volume = {77},
year = {2002},
doi = {10.1007/s00014-002-8341-3},
issn = {0010-2571},
} -
[Ch1] X. Chen, "The space of Kähler metrics," J. Differential Geom., vol. 56, iss. 2, pp. 189-234, 2000.
@article{Ch1,
author = {Chen, X.},
journal = {J. Differential Geom.},
number = {2},
pages = {189--234},
title = {The space of {K}ähler metrics},
volume = {56},
year = {2000},
issn = {0022-040X},
url = {http://projecteuclid.org/euclid.jdg/1090347643},
} -
[Ti2] G. Tian, "Extremal metrics and geometric stability," Houston J. Math., vol. 28, iss. 2, pp. 411-432, 2002.
@article{Ti2,
author = {Tian, Gang},
journal = {Houston J. Math.},
note = {special issue for S. S. Chern},
number = {2},
pages = {411--432},
title = {Extremal metrics and geometric stability},
volume = {28},
year = {2002},
issn = {0362-1588},
} -
[M1] T. Mabuchi, "Some symplectic geometry on compact Kähler manifolds. I," Osaka J. Math., vol. 24, iss. 2, pp. 227-252, 1987.
@article{M1,
author = {Mabuchi, Toshiki},
journal = {Osaka J. Math.},
number = {2},
pages = {227--252},
title = {Some symplectic geometry on compact {K}ähler manifolds. {I}},
volume = {24},
year = {1987},
issn = {0030-6126},
url = {http://projecteuclid.org/euclid.ojm/1200780161},
} -
[Se] S. Semmes, "Complex Monge-Ampère and symplectic manifolds," Amer. J. Math., vol. 114, iss. 3, pp. 495-550, 1992.
@article{Se,
author = {Semmes, Stephen},
journal = {Amer. J. Math.},
number = {3},
pages = {495--550},
title = {Complex {M}onge-{A}mpère and symplectic manifolds},
volume = {114},
year = {1992},
doi = {10.2307/2374768},
issn = {0002-9327},
} -
[D2] S. K. Donaldson, "Symmetric spaces, Kähler geometry and Hamiltonian dynamics," in Northern California Symplectic Geometry Seminar, Providence, RI: Amer. Math. Soc., 1999, vol. 196, pp. 13-33.
@incollection{D2, address = {Providence, RI},
author = {Donaldson, Simon K.},
booktitle = {Northern {C}alifornia {S}ymplectic {G}eometry {S}eminar},
pages = {13--33},
publisher = {Amer. Math. Soc.},
series = {Amer. Math. Soc. Transl. Ser. 2},
title = {Symmetric spaces, {K}ähler geometry and {H}amiltonian dynamics},
volume = {196},
year = {1999},
} -
[CC] E. Calabi and X. Chen, "The space of Kähler metrics II," J. Differential Geom., vol. 61, iss. 2, pp. 173-193, 2002.
@article{CC,
author = {Calabi, E. and Chen, X.},
journal = {J. Differential Geom.},
number = {2},
pages = {173--193},
title = {The space of {K}ähler metrics {II}},
volume = {61},
year = {2002},
issn = {0022-040X},
url = {http://projecteuclid.org/euclid.jdg/1090351383},
} -
[Ch3] X. Chen, "Space of Kähler metrics. III. On the lower bound of the Calabi energy and geodesic distance," Invent. Math., vol. 175, iss. 3, pp. 453-503, 2009.
@article{Ch3,
author = {Chen, X.},
journal = {Invent. Math.},
number = {3},
pages = {453--503},
title = {Space of {K}ähler metrics. {III}. {O}n the lower bound of the {C}alabi energy and geodesic distance},
volume = {175},
year = {2009},
doi = {10.1007/s00222-008-0153-7},
issn = {0020-9910},
} -
[Ch2] X. Chen, "On the lower bound of the Mabuchi energy and its application," Internat. Math. Res. Notices, iss. 12, pp. 607-623, 2000.
@article{Ch2,
author = {Chen, X.},
journal = {Internat. Math. Res. Notices},
number = {12},
pages = {607--623},
title = {On the lower bound of the {M}abuchi energy and its application},
year = {2000},
doi = {10.1155/S1073792800000337},
issn = {1073-7928},
} -
[CS] X. Chen and S. Sun, "Space of Kähler metrics (V) — Kähler quantization," in Metric and Differential Geometry. The Jeff Cheeger Volume, Boston: Birkhäuer, 2012, vol. 297, pp. 19-41.
@incollection{CS, address = {Boston},
author = {Chen, X. and Sun, S.},
booktitle = {Metric and Differential Geometry. The Jeff Cheeger Volume},
pages = {19--41},
publisher = {Birkhäuer},
series = {Progr. Math.},
title = {Space of {K}ähler metrics {(V)} --- {K}ähler quantization},
volume = {297},
year = {2012},
doi = {10.1007/978-3-0348-0257-4_2},
} -
[Be] B. Berndtsson, Probability measures related to geodesics in the space of Kähler metrics.
@misc{Be,
author = {Berndtsson, B.},
title = {Probability measures related to geodesics in the space of {K}ähler metrics},
} -
[DK] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, New York: The Clarendon Press, Oxford University Press, 1990.
@book{DK, address = {New York},
author = {Donaldson, Simon K. and Kronheimer, P. B.},
pages = {x+440},
publisher = {The Clarendon Press, Oxford University Press},
series = {Oxford Math. Monogr.},
title = {The Geometry of Four-Manifolds},
year = {1990},
isbn = {0-19-853553-8},
} -
[Le] E. Lerman, "Gradient flow of the norm squared of a moment map," Enseign. Math., vol. 51, iss. 1-2, pp. 117-127, 2005.
@article{Le,
author = {Lerman, Eugene},
journal = {Enseign. Math.},
number = {1-2},
pages = {117--127},
title = {Gradient flow of the norm squared of a moment map},
volume = {51},
year = {2005},
issn = {0013-8584},
} -
[Ke] G. R. Kempf, "Instability in invariant theory," Ann. of Math., vol. 108, iss. 2, pp. 299-316, 1978.
@article{Ke,
author = {Kempf, George R.},
journal = {Ann. of Math.},
number = {2},
pages = {299--316},
title = {Instability in invariant theory},
volume = {108},
year = {1978},
doi = {10.2307/1971168},
issn = {0003-486X},
} -
[Ne] L. Ness, "A stratification of the null cone via the moment map," Amer. J. Math., vol. 106, iss. 6, pp. 1281-1329, 1984.
@article{Ne,
author = {Ness, Linda},
journal = {Amer. J. Math.},
note = {with an appendix by David Mumford},
number = {6},
pages = {1281--1329},
title = {A stratification of the null cone via the moment map},
volume = {106},
year = {1984},
doi = {10.2307/2374395},
issn = {0002-9327},
} -
[Ki] F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton, NJ: Princeton Univ. Press, 1984, vol. 31.
@book{Ki, address = {Princeton, NJ},
author = {Kirwan, Frances Clare},
pages = {i+211},
publisher = {Princeton Univ. Press},
series = {Math. Notes},
title = {Cohomology of Quotients in Symplectic and Algebraic Geometry},
volume = {31},
year = {1984},
isbn = {0-691-08370-3},
} -
[GS] V. Guillemin and S. Sternberg, "A normal form for the moment map," in Differential Geometric Methods in Mathematical Physics, Dordrecht: Reidel, 1984, vol. 6, pp. 161-175.
@incollection{GS, address = {Dordrecht},
author = {Guillemin, Victor and Sternberg, Shlomo},
booktitle = {Differential Geometric Methods in Mathematical Physics},
pages = {161--175},
publisher = {Reidel},
series = {Math. Phys. Stud.},
title = {A normal form for the moment map},
volume = {6},
year = {1984},
} -
[OR] J. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Boston: Birkhäuser, 2004, vol. 222.
@book{OR, address = {Boston},
author = {Ortega, Juan-Pablo and Ratiu, Tudor S.},
pages = {xxxiv+497},
publisher = {Birkhäuser},
series = {Progr. Math.},
title = {Momentum Maps and {H}amiltonian Reduction},
volume = {222},
year = {2004},
isbn = {0-8176-4307-9},
} -
[CxH2] X. Chen and W. He, "The Calabi flow on Kähler surfaces with bounded Sobolev constant (I)," Math. Ann., vol. 354, iss. 1, pp. 227-261, 2012.
@article{CxH2,
author = {Chen, X. and He, Weiyong},
journal = {Math. Ann.},
number = {1},
pages = {227--261},
title = {The {C}alabi flow on {K}ähler surfaces with bounded {S}obolev constant ({I})},
volume = {354},
year = {2012},
doi = {10.1007/s00208-011-0723-7},
issn = {0025-5831},
} -
[Ca1] E. Calabi, "Extremal Kähler metrics," in Seminar on Differential Geometry, Yau, S. T., Ed., Princeton, N.J.: Princeton Univ. Press, 1982, vol. 102, pp. 259-290.
@incollection{Ca1, address = {Princeton, N.J.},
author = {Calabi, Eugenio},
booktitle = {Seminar on {D}ifferential {G}eometry},
editor = {Yau, S. T.},
pages = {259--290},
publisher = {Princeton Univ. Press},
series = {Ann. of Math. Stud.},
title = {Extremal {K}ähler metrics},
volume = {102},
year = {1982},
} -
[Ca2] E. Calabi, "Extremal Kähler metrics. II," in Differential Geometry and Complex Analysis, Chavel, I. and Farkas, H. M., Eds., New York: Springer-Verlag, 1985, pp. 95-114.
@incollection{Ca2, address = {New York},
author = {Calabi, Eugenio},
booktitle = {Differential Geometry and Complex Analysis},
editor = {Chavel, I. and Farkas, H. M.},
pages = {95--114},
publisher = {Springer-Verlag},
title = {Extremal {K}ähler metrics. {II}},
year = {1985},
} -
[CxH1] X. Chen and W. Y. He, "On the Calabi flow," Amer. J. Math., vol. 130, iss. 2, pp. 539-570, 2008.
@article{CxH1,
author = {Chen, X. and He, W. Y.},
journal = {Amer. J. Math.},
number = {2},
pages = {539--570},
title = {On the {C}alabi flow},
volume = {130},
year = {2008},
doi = {10.1353/ajm.2008.0018},
issn = {0002-9327},
} -
[CLW] X. Chen, H. Li, and B. Wang, "Kähler-Ricci flow with small initial energy," Geom. Funct. Anal., vol. 18, iss. 5, pp. 1525-1563, 2009.
@article{CLW,
author = {Chen, X. and Li, Haozhao and Wang, Bing},
journal = {Geom. Funct. Anal.},
number = {5},
pages = {1525--1563},
title = {Kähler-{R}icci flow with small initial energy},
volume = {18},
year = {2009},
doi = {10.1007/s00039-008-0690-7},
issn = {1016-443X},
} -
[TZ2] G. Tian and X. Zhu, Perelman’s $\mathcal{W}$-functional and stability of Kähler-Ricci flow.
@misc{TZ2,
author = {Tian, Gang and Zhu, Xiaohua},
title = {Perelman's {$\mathcal{W}$}-functional and stability of {K}ähler-{R}icci flow},
} -
[SW] S. Sun and Y-Q. Wang, On the Kähler-Ricci flow near a Kähler-Einstein metric.
@misc{SW,
author = {Sun, S. and Wang, Y-Q.},
journal = {J. Reine Angew. Math.},
note = {published online 2013-03-19},
title = {On the {K}ähler-{R}icci flow near a {K}ähler-{E}instein metric},
doi = {10.1515/crelle-2013-0004},
} -
[Si] L. Simon, "Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems," Ann. of Math., vol. 118, iss. 3, pp. 525-571, 1983.
@article{Si,
author = {Simon, Leon},
journal = {Ann. of Math.},
number = {3},
pages = {525--571},
title = {Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems},
volume = {118},
year = {1983},
doi = {10.2307/2006981},
issn = {0003-486X},
} -
[Ra] J. Råde, "On the Yang-Mills heat equation in two and three dimensions," J. Reine Angew. Math., vol. 431, pp. 123-163, 1992.
@article{Ra,
author = {R{\aa}de, Johan},
journal = {J. Reine Angew. Math.},
pages = {123--163},
title = {On the {Y}ang-{M}ills heat equation in two and three dimensions},
volume = {431},
year = {1992},
doi = {10.1515/crll.1992.431.123},
issn = {0075-4102},
} -
[D4] S. K. Donaldson, Floer Homology Groups in Yang-Mills Theory, Cambridge: Cambridge Univ. Press, 2002, vol. 147.
@book{D4, address = {Cambridge},
author = {Donaldson, Simon K.},
note = {with the assistance of M. Furuta and D. Kotschick},
pages = {viii+236},
publisher = {Cambridge Univ. Press},
series = {Cambridge Tracts in Math.},
title = {Floer Homology Groups in {Y}ang-{M}ills Theory},
volume = {147},
year = {2002},
doi = {10.1017/CBO9780511543098},
isbn = {0-521-80803-0},
} -
[FS] A. Fujiki and G. Schumacher, "The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics," Publ. Res. Inst. Math. Sci., vol. 26, iss. 1, pp. 101-183, 1990.
@article{FS,
author = {Fujiki, Akira and Schumacher, Georg},
journal = {Publ. Res. Inst. Math. Sci.},
number = {1},
pages = {101--183},
title = {The moduli space of extremal compact {K}ähler manifolds and generalized {W}eil-{P}etersson metrics},
volume = {26},
year = {1990},
doi = {10.2977/prims/1195171664},
issn = {0034-5318},
} -
[Ch4] X. Chen, The space of Kähler metrics (IV)—On the lower bound of the K energy.
@misc{Ch4,
author = {Chen, X.},
title = {The space of {K}ähler metrics {(IV)}---On the lower bound of the {K} energy},
} -
[Sz] G. Székelyhidi, "The Kähler-Ricci flow and $K$-polystability," Amer. J. Math., vol. 132, iss. 4, pp. 1077-1090, 2010.
@article{Sz,
author = {Sz{é}kelyhidi, G{á}bor},
journal = {Amer. J. Math.},
number = {4},
pages = {1077--1090},
title = {The {K}ähler-{R}icci flow and {$K$}-polystability},
volume = {132},
year = {2010},
doi = {10.1353/ajm.0.0128},
issn = {0002-9327},
} -
[To] V. Tosatti, "The K-energy on small deformations of constant scalar curvature Kähler manifolds," in Advances in Geometric Analysis, Somerville, MA: Int. Press, 2012, vol. 21, pp. 139-147.
@incollection{To, address = {Somerville, MA},
author = {Tosatti, Valentino},
booktitle = {Advances in Geometric Analysis},
pages = {139--147},
publisher = {Int. Press},
series = {Adv. Lect. Math. (ALM)},
title = {The {K}-energy on small deformations of constant scalar curvature {K}ähler manifolds},
volume = {21},
year = {2012},
} -
[Ku] M. Kuranishi, "New proof for the existence of locally complete families of complex structures," in Proc. Conf. Complex Analysis, New York: Springer-Verlag, 1965, pp. 142-154.
@incollection{Ku, address = {New York},
author = {Kuranishi, M.},
booktitle = {Proc. {C}onf. {C}omplex {A}nalysis},
pages = {142--154},
publisher = {Springer-Verlag},
title = {New proof for the existence of locally complete families of complex structures},
year = {1965},
} -
[FS2] A. Fujiki and G. Schumacher, "The moduli space of Kähler structures on a real compact symplectic manifold," Publ. Res. Inst. Math. Sci., vol. 24, iss. 1, pp. 141-168, 1988.
@article{FS2,
author = {Fujiki, Akira and Schumacher, Georg},
journal = {Publ. Res. Inst. Math. Sci.},
number = {1},
pages = {141--168},
title = {The moduli space of {K}ähler structures on a real compact symplectic manifold},
volume = {24},
year = {1988},
doi = {10.2977/prims/1195175329},
issn = {0034-5318},
} -
[Fi] J. Fine, "Calabi flow and projective embeddings," J. Differential Geom., vol. 84, iss. 3, pp. 489-523, 2010.
@article{Fi,
author = {Fine, Joel},
journal = {J. Differential Geom.},
number = {3},
pages = {489--523},
title = {Calabi flow and projective embeddings},
volume = {84},
year = {2010},
issn = {0022-040X},
url = {http://projecteuclid.org/euclid.jdg/1279114299},
} -
[OSY] H. Ono, Y. Sano, and N. Yotsutani, "An example of an asymptotically Chow unstable manifold with constant scalar curvature," Ann. Inst. Fourier $($Grenoble$)$, vol. 62, iss. 4, pp. 1265-1287, 2012.
@article{OSY,
author = {Ono, Hajime and Sano, Yuji and Yotsutani, Naoto},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
number = {4},
pages = {1265--1287},
title = {An example of an asymptotically {C}how unstable manifold with constant scalar curvature},
volume = {62},
year = {2012},
doi = {10.5802/aif.2722},
issn = {0373-0956},
} -
[DZ] A. Della Vedova and F. Zuddas, "Scalar curvature and asymptotic Chow stability of projective bundles and blowups," Trans. Amer. Math. Soc., vol. 364, iss. 12, pp. 6495-6511, 2012.
@article{DZ,
author = {Della Vedova, Alberto and Zuddas, Fabio},
journal = {Trans. Amer. Math. Soc.},
number = {12},
pages = {6495--6511},
title = {Scalar curvature and asymptotic {C}how stability of projective bundles and blowups},
volume = {364},
year = {2012},
doi = {10.1090/S0002-9947-2012-05587-5},
issn = {0002-9947},
} -
[APS] C. Arezzo, F. Pacard, and M. Singer, "Extremal metrics on blowups," Duke Math. J., vol. 157, iss. 1, pp. 1-51, 2011.
@article{APS,
author = {Arezzo, Claudio and Pacard, Frank and Singer, Michael},
journal = {Duke Math. J.},
number = {1},
pages = {1--51},
title = {Extremal metrics on blowups},
volume = {157},
year = {2011},
doi = {10.1215/00127094-2011-001},
issn = {0012-7094},
} -
[Ha] R. S. Hamilton, "The inverse function theorem of Nash and Moser," Bull. Amer. Math. Soc., vol. 7, iss. 1, pp. 65-222, 1982.
@article{Ha,
author = {Hamilton, Richard S.},
journal = {Bull. Amer. Math. Soc.},
number = {1},
pages = {65--222},
title = {The inverse function theorem of {N}ash and {M}oser},
volume = {7},
year = {1982},
doi = {10.1090/S0273-0979-1982-15004-2},
issn = {0273-0979},
} -
[We] A. Weinstein, "Symplectic manifolds and their Lagrangian submanifolds," Advances in Math., vol. 6, pp. 329-346 (1971), 1971.
@article{We,
author = {Weinstein, Alan},
journal = {Advances in Math.},
pages = {329--346 (1971)},
title = {Symplectic manifolds and their {L}agrangian submanifolds},
volume = {6},
year = {1971},
doi = {10.1016/0001-8708(71)90020-X},
issn = {0001-8708},
} -
[Ta] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991, vol. 100.
@book{Ta,
author = {Taylor, Michael E.},
pages = {213},
publisher = {Birkhäuser},
series = {Progr. Math.},
title = {Pseudodifferential Operators and Nonlinear {PDE}},
volume = {100},
year = {1991},
doi = {10.1007/978-1-4612-0431-2},
isbn = {0-8176-3595-5},
}
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