In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

Let $(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present talk is concerned with the set $\Red(z)$ of all reduced expressions for any $z\in W$. By associating each bc-expression to a certain symbol, we describe the set $\Red(z)$ and compute its cardinal $|\Red(z)|$ in terms of symbols. An explicit formula for $|\Red(z)|$ is deduced, where the Fibonacci numbers play a crucial role.

We investigate the fully nonlinear elliptic equations F(D^2u,Du,u,x) = 0, which satisfy the structural condition previously posed by Bianchini-Longinetti-Salani in 2009. By establishing a novel differential inequality, we prove a weak Harnack inequality for the principal curvatures of the level surfaces of the solutions. This result is indeed a quantitative version of the constant rank theorem showed by Guan-Xu in 2013.

In this talk, we will discuss stability and index estimates for compact and noncompact capillary surfaces. A classical result in minimal surface theory says that a stable complete minimal surface in R^3 must be a plane. We show that, under certain curvature assumptions, a strongly stable capillary surface in a 3-manifold with boundary has only three possible topological configurations. In particular, we prove that a strongly stable capillary surface in a half-space of R^3 which is minimal or has the contact angle less than or equal to $\pi/2$ must be a half-plane. We also give index estimates for compact capillary surfaces in 3-manifolds by using harmonic one-forms.

Weyl laws relate the asymptotic behaviors of the eigenvalues of certain geometric operators with the geometric/dynamical properties of the underlying space. In this talk I will briefly describe these connections, with an emphasis on the relation between the eigenvalue counting problem for special planar domains with integrable billiard flows and the classical lattice point counting problem.

We will introduce the S-closed conformal transformation in Finsler geometry. We show some properties of such transformation. Using some results, we show how to refine a rigidity theorem of Matsumoto problem about conformally equivalent between two non-Riemannian Berwald manifolds.

Q-curvature is 4th-order analog of scalar curvature, which is an important geometric object in the study of conformal geometry. In this talk, after reviewing some of our previous work about deformations of Q-curvature, I will talk about some new progresses in the study of the volume comparison result of Einstein manifolds with respect to Q-curvature. This series of works are joint with Yueh-Ju Lin in Wichita State University.

Let Ω be a domain of a closed manifold $(M, g_0)$ with dim M>2. Let $g=u^\frac{4}{n-2}g_0$ be a complete metric defined on Ω. We will show that $M\setminus Ω$ is a finite set when $\int_Ω|Ric(g)|^\frac{n}{2}dV_g<+∞. Such a result is not true if we replace Ricci curvature with Scalar curvature. We will discuss the properties of conformal metrics with $\|R\|_{L^\frac{n}{2}}<+∞$ on a punctured ball of a Riemannian manifold , and give some geometric obstacles for Huber's theorem in this case.