The local equivalence problem in CR geometry

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape Analysis and Conformal Equivalence.

On the equivalence between local and global existence of complete Kähler metrics with plurisubharmonic potentials

Like the classical potential theory, it was conjectured that there exists equivalence between locally and globally pluripolar and complete pluripolar sets, namely, Problem I of Lelong, and was solved by Josefson, Bedford - Taylor and Colţoiu. In this article, we consider complements of complete Kähler domains as the generalization of closed complete pluripolar sets and prove that there exists an equivalence between local and global existence of these sets.

Normal forms in Cauchy-Riemann Geometry: a survey

One of effective ways to solve the equivalence problem and describe moduli spaces for real submanifolds in complex space is the normal form approach. In this survey, we outline some normal form constructions in CR-geometry and formulate a number of open problems.

Local equivalence problem for Levi flat hypersurfaces

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S_1 ~ S_2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant for this relation, which allows to prove some characterizations of triviality (i.e. equivalence to the hyperplane). Then, we employ the same invariant to construct infinitely many non-trivial classes, including an infinite family of not equivalent hypersurfaces which are almost everywhere analytic.

Bounded differentials on unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to $r$-differetials. We study the relationship between bounded holomorphic $r$-differentials and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space $SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles. Also, we show the equivalences between the boundedness of holomorphic differentials and havi…

广义Kähler-Ricci孤立子的环面几何

Toric geometry of generalized Kähler-Ricci solitons

该研究建立环面稳态Kähler-Ricci孤立子与A型环面GKRS的局部及全局等价关系，给出高维新例构造方法，并完成四维GKRS的局部分类与完备辛型例子的推测性穷尽构造。