arXiv subjects

Rong Du

Publications and source records attributed to Rong Du.

44 records · Page 3Linked to original sources

arXiv2016

On higher dimensional complex Plateau problem

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn--Rossi cohomology groups on $X$ in 1981. For $n=2$ and $N\ge n+1$, the first and third authors introduced a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. For $n\ge 3$ and $N>n+1$, the problem still remains open. In this paper, we generalize the invariant $g^{(1,1)}(X)$ to higher dimension as $g^{(Λ^n 1)}(X)$ and show that if $g^{(Λ^n 1)}(X)=0$, then the interior has at most finite number of rational singularities. In particular, if $X$ is Calabi--Yau of real dimension $5$, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization.

math.AG↗

arXiv2016

On abelian canonical n-folds of general type

Let $X$ be a Gorenstein minimal projective $n$-fold with at worst locally factorial terminal singularities, and suppose that the canonical map of $X$ is generically finite onto its image. When $n<4$, the canonical degree is universally bounded. While the possibility of obtaining a universal bound on the canonical degree of $X$ for $n \geqslant 4$ may be inaccessible, we give a uniform upper bound for the degrees of certain abelian covers. In particular, we show that if the canonical divisor $K_X$ defines an abelian cover over $\mathbb{P}^n$, i.e., when $X$ is an \emph{abelian canonical $n$-fold}, then the canonical degree of $X$ is universally upper bounded by a constant which only depends on $n$ for $X$ non-singular. We also construct two examples of non-singular minimal projective $4$-folds of general type with canonical degrees $81$ and $128$.

math.AG↗

arXiv2016

On the canonical degrees of Gorenstein threefolds of general type

Let $X$ be a Gorenstein minimal projective $3$-fold with at worst locally factorial terminal singularities. Suppose that the canonical map is generically finite onto its image. C. Hacon showed that the canonical degree is universally bounded by $576$. We improved Hacon's universal bound to $360$. Moreover, we gave all the possible canonical degrees of $X$ if $X$ is an abelian cover over $\mathbb{P}^3$ and constructed all the examples with these canonical degrees.

math.AG↗

arXiv2015

Application of Mythen Detector In-situ XRD Study on The Thermal Expansion Behavior of Metal Indium

A Mythen detector has been equipped at the beamline 4B9A of Beijing Synchrotron Radiation Facility, which can be used for in-situ real-time measurement of X-ray diffraction (XRD) full profiles. In this paper, the thermal expansion behavior of metal indium has been studied by using the in-situ XRD technique with the Mythen detector. The indium film was heated from 30 to 160 °C with a heating rate of 2 °C/min. The in-situ XRD full-profiles were collected with a rate of one profile per 10 seconds. Rietveld refinement was used to extract the structural parameters. The results demonstrate that the thermal expansion of metal indium is nonlinear especially when the sample temperature was close to its melting point (156.5 °C). The expansion of a-axis and the contraction of c-axis of the tetragonal unit cell of metallic indium can be well described by biquadratic and cubic polynomials, respectively. The tetragonal unit cell presents a tendency to become cubic one with the increase of temperature but without detectable phase change. This study is not only beneficial to the application of metal indium, but also exhibits the capacity of in-situ time-resolved XRD experiments at the X-ray diffraction station of BSRF.

cond-mat.mtrl-sci↗

arXiv2014

Canonical maps of surfaces defined by Abelian covers

In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $\mathbb{Z}_2^{\oplus 3}$ and give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.

math.AG↗

arXiv2014

On the Griffiths numbers for higher dimensional singularities

We show that Yau's conjecture on the inequalities for (n-1)-th Griffiths number and (n-1)-th Hironaka number does not hold for isolated rigid Gorenstein singularities of dimension greater than 2. But his conjecture on the inequality for (n-1)-th Griffiths number is true for irregular singularities.

math.AG↗

arXiv2012

Hierarchical structure and biomineralization in cricket tooth

Cricket is a truculent insect with stiff and sharp teeth as a fighting weapon. The structure and possible biomineralization of the cricket teeth are always interested. Synchrotron radiation X-ray fluorescence, X-ray diffraction and small angle X-ray scattering techniques were used to probe the element distribution, possible crystalline structures and size distribution of scatterers in cricket teeth. Scanning electron microscope was used to observe the nanoscaled structure. The results demonstrate that Zn is the main heavy element in cricket teeth. The surface of the cricket teeth has a crystalline compound like ZnFe2(AsO4)2(OH)2(H2O)4. While, the interior of the teeth has a crystalline compound like ZnCl2, which is from the biomineralization. The ZnCl2-like biomineral forms nanoscaled microfibrils and their axial direction points at the top of tooth cusp. The microfibrils aggregate random into intermediate filaments, forming a hierarchical structure. A sketch map of the cricket tooth cusp was proposed and a detailed discussion was given in this paper.

physics.bio-ph↗

arXiv2012

Kohn-Rossi Cohomology and its application to the Complex Plateau Problem III

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension 2n-1 in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn-Rossi cohomology groups on $X$ in 1981. For n=2 and $N\ge n+1$, the problem has been open for over 30 years. In this paper we introduce a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. In case $n=2$ and N=3, the vanishing of this invariant is enough to give the interior regularity.

math.DG↗