The regularity of harmonic maps into spheres and applications to Bernstein problems

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial

Gauge choices and Entanglement Entropy of two dimensional lattice gauge fields

In this paper, we explore the question of how different gauge choices in a gauge theory affect the tensor product structure of the Hilbert space in configuration space. In particular, we study the Coulomb gauge and observe that the naive gauge potential degrees of freedom cease to be local operators as soon as we impose the Dirac brackets. We construct new local set of operators and compute the entanglement entropy according to this algebra in $2+1$ dimensions. We find that our proposal would lead to an entanglement entropy that behave very similar to a single scalar degree of freedom if we do not include further centers, but approaches that of a gauge field if we include non-trivial centers. We explore also the situation where the gauge field is Higgsed, and construct a local operator algebra that again requires some deformation. This should give us some insight into interpreting the entanglement entropy in generic gauge theories and perhaps also in gravitational theories.Comment: 38 pages,25 figure

A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map $(\phi,\psi)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|\psi|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|\psi|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-harmonic maps from $S^2$ to $S^2$.Comment: 12 page

Injectivity radius for non-simply connected symmetric spaces via Cartan polyhedron

We determine the cut locus of arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.Comment: 25 page

Power, identity and antiquarian approaches in modern Chinese art

The pursuit of antiquity was important for scholarly artists in constructing their knowledge of history and cultural identity in late Imperial China. Following various publications by Bi Yuan 畢沅 (1730-1797), Wu Yi 武億 (1745-1799) and Qian Daxin 錢大昕 (1728-1804) in the 18th century, the study and collecting of rubbings of Northern Wei stone inscriptions and steles was popular. Such spread of interest in jinshi, inscriptions on metal and stone, also formed a base for studying seal carving, epigraphy and archaic painting. While traditional antiquarians would cherish inscriptions which enabled them to correct mistakes in the transmitted historical texts and the Classics, however, much of the antiquarian activity was adapted to mere literary exercise or connoisseurship, for instance, to supplying materials which could provide models for seal-carving and calligraphy. Examples could be seen in the calligraphy works and seal carvings of the Xiling bajia 西泠八家 (Eight Masters of Xiling, i.e. Hangzhou), also known as Zhe School of Calligraphy and Carving. Their keen interest in seeking inspiration from steles for their artistic presentations has been recorded in their writing and painting. In addition, the way the scholar-collector of the 19th and early 20th centuries mounted the rubbings, seals, inscriptions, paintings, letters and textual evidence studies into one album shows a changing ideology: rubbings were not only for scholarly study in classical learning, but were regarded as part of the art form and were appreciated on various social occasions. The antiquarian movement ultimately served as a tool for re-writing art historiography in modern China. This paper aims to address the phenomenon and formation of the jinshi painting that dominated in late Imperial and early modern China. Through case studies of three important jinshi societies in Shanghai, I will investigate in what way literary taste from the southern region gradually replaced imperial patronage which was in decline after the Qianlong emperor’s reign, and how the shift of the cultural centre from Beijing to the southern regions from the mid-19th century onwards became a reflection of changing power and identity for cultural leaders and their perspectives in history and the history of objects