Guofang Wang

Existence results for mean field equations

Abstract Let Ω be an annulus. We prove that the mean field equation −δψ= e −βψ ∫ Ω e−βψ in Ω ψ=0 on ∂Ω admits a solution for β ∈ (−16π, −8π). This is a supercritical case for the Moser-Trudinger inequality.

Some properties of the Schouten tensor and applications to conformal geometry

The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the kth elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ + k . We prove that this eigenvalue condition for k > n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.

Classification of solutions of a Toda system in R2

We consider solutions of a Toda system for SU(N+1) and show that any solution with finite exponential integral cam be obtained from a rational curve in complex projective space of dimension N

Liouville theorems for Dirac-harmonic maps

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space Rn, the hyperbolic space Hn, and a Riemannian manifold Sn (n⩾3) with the Schwarzschild metric to any Riemannian manifold N.

Entanglement of Formation for a Class of Quantum States

Abstract Entanglement of formation for a class of higher-dimensional quantum mixed states is studied in terms of a generalized formula of concurrence for N -dimensional quantum systems. As applications, the entanglement of formation for a class of 16×16 density matrices are calculated.

Variational aspects of the Seiberg-Witten functional

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection A on a line bundleL and a sectionφ of another bundleW+ constructed fromL and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.

The boundary value problem for Dirac-harmonic maps

Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We show that a weakly Diracharmonic map is smooth in the interior of the domain. We also prove regularity results for Dirac-harmonic maps at the boundary when they solve an appropriate boundary value problem which is the mathematical interpretation of the D-branes of superstring theory.

Steady state solutions of a reaction-diffusion system modeling chemotaxis

We study the following nonlinear elliptic equation where Ω is a smooth bounded domain in ℝ2. This equation arises in the study of stationary solutions of a chemotaxis system proposed by Keller and Segel. Under the condition that for m =1,2,…, where λ1 is the first (nonzero) eigenvalue of —Δ under the Neumann boundary condition, we establish the existence of a solution to the above equation. Our idea is a combination of Struwes technique and blow up analysis for a problem with Neumann boundary condition.

An almost Schur theorem on 4-dimensional manifolds

In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

The Sasaki-Ricci flow

In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kahler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Caos results on the Kahler–Ricci flow.