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Etnyre J B, Lafountain D J, Tosun B, et al. Legendrian and transverse cables of positive torus knots[J]. Geometry \u0026 Topology, 2011, 16(3): 1639-1689.
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Andy Wand. Factorizations of diffeomorphisms of compact surfaces with boundary. 2009.
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Etnyre J B, Van Hornmorris J. Monoids in the mapping class group[J]. Geometry and Topology Monographs, 2015, 19(1): 319-365.
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Li Y, Wang J. The support genus of certain Legendrian knots[J]. Journal of Knot Theory and Its Ramifications, 2011, 21(11).
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Kai Cieliebak · Evgeny Volkov. Stable Hamiltonian structures in dimension three are supported by open books. 2010.
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Paolo Lisca. Stein fillable contact 3–manifolds and positive open books of genus one. 2014.
In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary
to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies
cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface
that have contact geometric significance.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger–Gromoll splitting theorem and Cheng’s maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
Hutchings M. Quantitative embedded contact homology[J]. Journal of Differential Geometry, 2010, 88(2): 231-266.
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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.
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Richard Hind · Samuel T Lisi. Symplectic embeddings of polydisks. 2013.
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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.
We prove a sharp inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general $n \ge 3$. This inequality generalizes the classical Minkowski inequality [19] for surfaces in the three dimensional Euclidean space. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established in [4].
Huai-Dong CaoDepartment of Mathematics, Lehigh University, Bethlehem, PA 18015, USAXiaofeng SunDepartment of Mathematics, Lehigh University, Bethlehem, PA 18015, USAShing-Tung YauDepartment of Mathematics, Harvard University, Cambridge, MA 02138, USAYingying ZhangYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
Differential Geometrymathscidoc:2204.10009
Journal of the Iranian Mathematical Society, 1, (1), 1-12, 2020.6
In this paper we survey certain aspects of the classical Weil-Petersson metric and its generalizations. Being a natural L^2 metric on the parameter space of a family of complex manifolds or holomorphic vector bundles which admit some canonical metrics, the Weil-Petersson metric is well defined when the automorphism group of each fiber is discrete and the curvature of the Weil-Petersson metric can be computed via certain integrals over each fiber. We will discuss the case when these fibers have continuous automorphism groups. We also discuss the relation between the Weil-Petersson metric and energy of harmonic maps.