Using the degeneration formula for Donaldson-Thomas invariants [L-W,MNOP2], we proved formulae for blowing up a point, simple flops, and extremal transitions.
Hutchings M. Quantitative embedded contact homology[J]. Journal of Differential Geometry, 2010, 88(2): 231-266.
2
Mcduff D, Schlenk F. The embedding capacity of 4-dimensional symplectic ellipsoids[J]. Annals of Mathematics, 2009, 175(3): 1191-1282.
3
Hutchings M. Recent progress on symplectic embedding problems in four dimensions[J]. Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(20): 8093-8099.
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Hutchings M. Lecture Notes on Embedded Contact Homology[C]., 2013: 389-484.
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Olguta Buse · Richard Hind. Symplectic embeddings of ellipsoids in dimension greater than four. 2011.
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Daniel Cristofarogardiner · Michael Hutchings · Vinicius Gripp Barros Ramos. The asymptotics of ECH capacities. 2012.
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Alberto Abbondandolo · Rostislav Matveyev. How large is the shadow of a symplectic ball. 2012.
8
Richard Hind · Samuel T Lisi. Symplectic embeddings of polydisks. 2013.
9
Mcduff D, Opshtein E. Nongeneric J–holomorphic curves and singular inflation[J]. Algebraic \u0026 Geometric Topology, 2013, 15(1): 231-286.
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Choi K, Cristofarogardiner D, Frenkel D, et al. Symplectic embeddings into four-dimensional concave toric domains[J]. Journal of Topology, 2013, 7(4): 1054-1076.
In this note we show that one open 4-dimensional ellipsoid embeds symplectically into another if and only if the ECH capacities
of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of
the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by
Hutchings’ recent results on embedded contact homology (ECH) capacities but does not use them.
Kwokwai ChanThe Chinese University of Hong KongCheol-Hyun ChoSeoul National UniversitySiu-Cheong LauHarvard UniversityHsian-Hua TsengOhio State University
Ke H, Zhou J. Quantum McKay correspondence for disc invariants of toric Calabi-Yau 3-orbifolds[J]. Acta Mathematica Sinica, 2014, 31(1): 29-34.
2
Lau S. Open Gromov-Witten invariants and SYZ under local conifold transitions[J]. Journal of The London Mathematical Society-second Series, 2013, 90(2): 413-435.
3
Huazhong Ke · Jian Zhou. Gauged Linear Sigma Model for Disc Invariants. 2014.
4
Cheolhyun Cho · Hansol Hong · Sanghyun Kim · Siucheong Lau. Lagrangian Floer potential of orbifold spheres. 2014.
5
Kwokwai Chan. The Strominger-Yau-Zaslow conjecture and its impact. 2014.
Chruściel P T, Costa J L, Heusler M, et al. Stationary Black Holes: Uniqueness and Beyond[J]. Living Reviews in Relativity, 2012.
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Chruściel P T, Eckstein M, Nguyen L, et al. Existence of singularities in two-Kerr black holes[J]. Classical and Quantum Gravity, 2011, 28(24).
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I P Costa E Silva. On the geodesic incompleteness of spacetimes containing marginally outer trapped surfaces. 2012.
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Eichmair M. The Jang Equation Reduction of the Spacetime Positive Energy Theorem in Dimensions Less Than Eight[J]. Communications in Mathematical Physics, 2012, 319(3): 575-593.
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Carlotto A. Rigidity of stable marginally outer trapped surfaces in initial data sets[J]. Annales Henri Poincaré, 2014, 17(10): 2825-2847.
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Lars Andersson · Mattias Dahl · Gregory J Galloway · Daniel Pollack. On the geometry and topology of initial data sets with horizons. 2015.
We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method
and tools from geometric measure theory to force and control a blow-up of Jang’s equation. Substantial new geometric insights
regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in
this paper are flexible and can be adapted to other non-variational existence problems.
Brendle S. Minimal surfaces in S^3: a survey of recent results[C]., 2013, 3(1): 133-171.
2
Heller S. A spectral curve approach to Lawson symmetric CMC surfaces of genus 2[J]. Mathematische Annalen, 2012: 607-652.
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Simon Brendle. Embedded Weingarten tori in S^3. 2013.
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Bettiol R G, Piccione P. Delaunay type hypersurfaces in cohomogeneity one manifolds[J]. International Mathematics Research Notices, 2013, 2016(10): 3124-3162.
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Hauswirth L, Kilian M, Schmidt M U, et al. The geometry of embedded constant mean curvature tori in the 3-sphere via integrable systems[C]., 2013.
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Heller L, Heller S, Schmitt N, et al. The spectral curve theory for $(k,l)-$symmetric CMC surfaces[J]. Journal of Geometry and Physics, 2015: 201-213.
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Giovanni Catino. A remark on compact hypersurfaces with constant mean curvature in space forms. 2014.
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Laurent Hauswirth · M Kilian · M U Schmidt. MEAN-CONVEX ALEXANDROV EMBEDDED CONSTANT MEAN CURVATURE TORI IN THE 3-SPHERE. 2013.
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Alencar H, Carmo M P. Hypersurfaces With Constant Mean Curvature in Spheres[J]. Proceedings of the American Mathematical Society, 1994, 120(4): 1223-1229.
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Michael T Anderson. Alexandrov immersions, holonomy and minimal surfaces in $S^3$. 2014.
We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.