A HARNACK TYPE INEQUALITY FOR SOME CONFORMALLY INVARIANT EQUATIONS ON HALF EUCLIDEAN SPACE

We establish a Harnack type inequality on half Euclidean space for general conformally invariant fully nonlinear elliptic equations of second order.

STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form △2ψ ＋ L（x,△ ψ）, including the conformal hessian operator.

REVISITING A NON-DEGENERACY PROPERTY FOR EXTREMAL MAPPINGS

We extend an earlier result obtained by the author in [7].

Gluing Affine Vortices

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition.This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model,such as the closed or open quantum Kirwan map.

Dual mean Minkowski measures of symmetry for convex bodies

We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which is in a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well.

Isolated complex singularities and their CR links Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type Ⅰ

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of Type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles,in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups.For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.

On the C^1 Regularity of Solutions to Divergence Form Elliptic Systems with Dini-Continuous Coefficients(Dedicated to Haim Brezis on his 70th birthday with friendship and admiration)

The author proves C^1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients.

Smoothness of the Gradient of Weak Solutions of Degenerate Linear Equations

Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.

Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R^(n).When the distance of inclusions,denoted byε,goes to 0,the gradient of solutions may blow up.When two inclusions are strictly convex,it was known that an upper bound of the blow-up rate is of orderε^(-1/2)for n=2,and is of orderε^(-1/2+β)for someβ>0 when dimension n≥3.In this paper,we generalize the above results for insulators with flatter boundaries near touching points.

Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity

Asymptotic expansions of the voltage potential in terms of the "radius" of a diametrically small(or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a "simple" replacement for the background potential, with the following properties:(1) This replacement may be(simply) calculated from the limiting domain Ω\σ, the boundary data on the boundary of Ω, and the right-hand side.(2) This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.(3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.

The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties with discrete automorphism groups.We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments.This perturbation approach depends on the valuative criterion and non-Archimedean estimates,and is motivated by our previous paper.

Dedication on the Occasion of Professor Xiaqi Ding's 80th Birthday

This volume is dedicated to Professor Xiaqi Ding, an academician of the Chinese Academy of Sciences, on the occasion of his eightieth birthday.

Sifting for small primes from an arithmetic progression

In this work and its sister paper(Friedlander and Iwaniec(2023)),we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression.Using sieve machinery in both papers,we are able to dispensh with the log-free zero density bounds and the repulsion property of exceptional zeros,two deep innovations begun by Linnik and relied on in earlier proofs.

Ricci曲率以及度量球局部回卷体积有下界的Riemann流形

本文是一篇关于Ricci曲率与固定半径度量球的局部回卷体积有下界流形M的近期研究综述.所谓在点x∈M处的度量球B_r(x)的局部回卷体积,是指B_r(x)的(不完备的)万有覆叠上以基本点为中心、r为半径的度量球体积.

A generalized π2-diffeomorphism finiteness theorem

Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n,and upper bounds on the absolute value of sectional curvature and diameter of M.In this paper,we will generalize thisπ2-diffeomorphism finiteness by removing the condition thatπ1(M)-0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M.

Survey on Discrete Surface Ricci Flow

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a nonlinear heat diffusion process, and becomes constant eventually. Ricci flow is a powerful computational tool to design Riemannian metrics by prescribed curvatures This work surveys the theory of discrete surface Ricci flow registration and shape analysis. Surface Ricci flow has been generalized to the discrete setting. its computational algorithms, and the applications for surface

Gradient estimates for parabolic systems from composite material

In this paper, we derive W^(1,∞) and piecewise C^(1,α) estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise H¨older continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56:892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.

Geometric and analytic problems for a real submanifold in C^n with CR singularities

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C^3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).

Monotonicity for the Chern-Moser-Weyl curvature tensor and CR embeddings

We give,in this paper,a monotonicity formula for the Chern-Moser-Weyl curvature tensor under the action of holomorphic embeddings between Levi non-degenerate hypersurfaces with the same positive signature.As an application,we provide some concrete examples of algebraic Levi non-degenerate hypersurfaces with positive signature that are not embeddable into a hyperquadric of the same signature in a complex space of higher dimension.