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We consider Carleson-Sj\"{o}lin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of H\"{o}rmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sj\"{o}lin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to H\"{o}rmander-type oscillatory integral operators, prove that if a H\"{o}rmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures., Comment: fixed typos

We proved a rigidity result for Delaunay triangulations of the plane under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We followed Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and developed a discrete Schwarz lemma and a discrete Liouville theorem. The main tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry., Comment: Final version, to appear in Advances in Mathematics

Photocatalytic water splitting is an effective way to obtain renewable clean energy. The challenge is to design tunable photocatalyst to meet the needs in different environments. At the same time, the oxygen and hydrogen evolution reactions (OER and HER) on the photocatalyst should be separated, which will be conducive to the separation of products. The electronic, optical and photocatalytic properties of Janus X2PAs (X = Si, Ge and Sn) monolayers are explored by first-principles calculation. All the strain-free X2PAs monolayers exhibit excellent photocatalytic properties with suitable band edge positions straddling the standard redox potential of water and large visible light absorption coefficients (up to 105 cm-1). Interestingly, the intrinsic internal electric field is favorable for separating photogenerated carriers to different surfaces of the monolayer. It contributes to realize the OER and HER separated on different sides of the monolayer. In particular, the energy band edge positions of X2PAs monolayers can be well adjusted by biaxial strain. Then it can effectively modulate photocatalytic reactions, suggesting X2PAs monolayers can be a piezo-photocatalytic switch between the OER, HER and full-reaction of redox for water splitting. This investigation not only highlights that the photocatalyst X2PAs monolayers with the separated OER and HER can be effectively tuned by the mechanical strain, but also provides a new strategy for designing highly adaptable and tunable piezo-photocatalysts., Comment: 22 pages, 5 figures

For a harmonic diffeomorphism between the Poincar\'{e} 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 having a negative upper bound of the induced curvature on hyperbolic affine spheres in $\mathbb{R}^3$, maximal surfaces in $\mathbb{H}^{2,n}$ and $J$-holomorphic curves in $\mathbb{H}^{4,2}$ respectively. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods., Comment: 34 pages. Comments are very welcome

In \cite{Wu22}, under the uniformly acute condition, Wu showed the rigidity of the geodesic triangulated plane under Luo's discrete conformality. In this article, by modifying Wu's proof, we improve this result by weakening the uniformly acute condition to the uniformly Delaunay condition., Comment: 9 pages, 1 figure

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits., Comment: 41 pages, comments are welcome

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of $f$. As an application, we give a complete picture for maximal $Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the Hitchin components., Comment: 27 pages, comments are welcome

Given a reductive representation $\rho: \pi_1(S)\rightarrow G$, there exists a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n-1}$ case. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system., Comment: 24 pages, revised version, to appear in J. Differ. Geom

In recent years, Streets and Tian introduced a series of curvature flows to study non-K\"{a}hler geometry. In this paper, we study how to construct second order curvature flows in a uniform way, under some natural assumptions which holds in Streets and Tian's works. As a result, by classifying the lower order tensors, we classify the second order curvature flows in almost Hermitian, almost K\"{a}hler and Hermitian geometries in certain sense., Comment: To appear in Annals of Global Analysis and Geometry