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Andy Wand. Factorizations of diffeomorphisms of compact surfaces with boundary. 2009.
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Etnyre J B, Li Y. The arc complex and contact geometry: non-destabilizable planar open book decompositions of the tight contact 3-sphere[J]. International Mathematics Research Notices, 2013, 2015(5): 1401-1420.
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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.
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.
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.
By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We
prove rigidity results for the squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.
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].