Hopf-type theorem for self-shrinkers

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

Online 3D model reconstruction and virtual display for garment

An online garment model reconstruction method is proposed using the axial deformation to simulate the various garment models to display the wearing effect under different sizes. The garment model is classified into three categories according to the structural features. The main sections and key points are determined based on the existing dimensional features. Inverse operation and radial weight are used in the garment deformation. Texture mapping is applied to the model reconstruction result to obtain a new model with clothing patterns. An online system is developed and garment models with multiple scales are applied to adapt to various devices. The reconstruction results are compared and analyzed, which proves that the method proposed can realize real-time 3D reconstruction of garment models with the changing sizes on the web page.

Radial symmetry of minimizers to the weighted Dirichlet energy

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

Hankel operators induced by radial Bekollé–Bonami weights on Bergman spaces

We study big Hankel operators $$H_f^\nu :A^p_\omega \rightarrow L^q_\nu $$ H f ν : A ω p → L ν q generated by radial Bekollé–Bonami weights $$\nu $$ ν , when $$1<p\le q<\infty $$ 1 < p ≤ q < ∞ . Here the radial weight $$\omega $$ ω is assumed to satisfy a two-sided doubling condition, and $$A^p_\omega $$ A ω p denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $$H_f^\nu $$ H f ν and $$H_{{\overline{f}}}^\nu $$ H f ¯ ν is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau et al. (Indiana Univ Math J 65(5):1639–1673, 2016), the respective spaces depend on the weights $$\omega $$ ω and $$\nu $$ ν in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.

GAMA+KiDS: Alignment of galaxies in galaxy groups and its dependence on galaxy scale

Intrinsic galaxy alignments are a source of bias for weak lensing measurements as well as a tool for understanding galaxy formation and evolution. In this work, we measure the alignment of shapes of satellite galaxies, in galaxy groups, with respect to the brightest group galaxy (BGG), as well as alignments of the BGG shape with the satellite positions, using the highly complete Galaxy And Mass Assembly (GAMA) spectroscopic survey and deep imaging from the Kilo Degree Survey. We control systematic errors with dedicated image simulations and measure accurate shapes using the DEIMOS shape measurement method. We find a significant satellite radial alignment signal, which vanishes at large separations from the BGG. We do not identify any strong trends of the signal with galaxy absolute magnitude or group mass. The alignment signal is dominated by red satellites. We also find that the outer regions of galaxies are aligned more strongly than their inner regions, by varying the radial weight employed during the shape measurement process. This behaviour is evident for both red and blue satellites. BGGs are also found to be aligned with satellite positions, with this alignment being stronger when considering the innermost satellites, using red BGGs and the shape of the outer region of the BGG. Lastly, we measure the global intrinsic alignment signal in the GAMA sample for two different radial weight functions and find no significant difference.

Radial averaging operator acting on Bergman and Lebesgue spaces

It is shown that the radial averaging operator T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)% \,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds, induced by a radial weight ω on the unit disc {\mathbb{D}} , is bounded from the weighted Bergman space {A^{p}_{\nu}} , where {0<p<\infty} and the radial weight ν satisfies {\widehat{\nu}(r)\leq C\widehat{\nu}(\frac{1+r}{2})} for all {0\leq r<1} , to {L^{p}_{\nu}} if and only if the self-improving condition \sup_{0\leq r<1}\frac{\widehat{\omega}(r)^{p}}{\int_{r}^{1}s\nu(s)\,ds}\int_{0% }^{r}\frac{t\nu(t)}{\widehat{\omega}(t)^{p}}\,dt<\infty is satisfied. Further, two characterizations of the weak-type inequality \eta(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f% \|_{L^{p}_{\nu}}^{p},\quad\lambda>0, are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases {A^{p}_{\nu}\to L^{p}_{\nu}} , {L^{p}_{\nu}\to L^{p}_{\nu}} and {L^{p}_{\nu}\to L^{p,\infty}_{\nu}} are analyzed.