GBU Geometric Analysis and PDE Seminar

host: Xumin Jiang, Zhehui Wang（王哲辉， wangzhehui@gbu.edu.cn）

Title: On Brezis' First Open Problem: A Complete Solution

Abstract: In 2023, H. Brezis published a list of his “favorite open problems”, which he described as challenges he had “raised throughout his career and has resisted so far”. We provide a complete resolution to the first one–Open Problem 1.1–in Brezis’s favorite open problems list: the existence of solutions to the long-standing Brezis-Nirenberg problem on a three-dimensional ball. Furthermore, using the building blocks of Del Pino-Musso-PacardPistoia sign-changing solutions to the Yamabe problem, we establish the existence of infinitely many sign-changing, nonradial solutions for the full range of the parameter.



Title: A priori interior estimates for special Lagrangian curvature

Abstract: We establish a priori interior curvature estimates for the special Lagrangian curvature equations in both the critical phase and convex cases. The supercritical case, however, is distinct from the special Lagrangian equations. In dimension two, we observe that this curvature equation is equivalent to the equation arising in the optimal transportation problem with a "relative heat cost" function, as discussed in Brenier's paper. When 0 < Θ < π/2 (supercritical phase), the equation violates the Ma-Trudinger-Wang condition. However, Loeper's counterexample for general optimal transport problems does not directly apply here, as this concerns a specific optimal transport problem with fixed density functions. Moreover, the interior gradient estimates for this curvature equation are simpler than those for the special Lagrangian equations. We have demonstrated that these gradient estimates also hold for subcritical phases. It is worth noting that for the special Lagrangian equation, particularly in subcritical phases, the interior gradient estimate remains an open problem. This is joint work with Xingchen Zhou.



Title: Splitting theorem in Kähler geometry

Abstract: In this talk, we focus on the relationship between the curvature and the distance on Riemannian manifolds. First we review classical results like the Bonnet-Myers theorem and the Cheeger-Gromoll splitting theorem. Then we introduce the mixed curvature on Kähler manifolds and present our recent splitting theorem for Kähler manifolds with nonnegative mixed curvature. If time permits, I'll mention possible extensions of our work in comparison to known results for Ricci curvature.



Title: The Prescribed Q-Curvature Flow for Even Dimension in a Critical Case

Abstract: The prescribed Q-curvature flow equation on a even dimensional closed Riemannian manifold (M,g), was introduced by S. Brendle in 2003, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and $\int_M Qd\mu<(n-1)!\Vol\left(S^n \right)$. In this talk I focus on the critical case that $\int_M Qd\mu=(n-1)!\Vol\left( S^n \right)$, I will show the convergence of the flow under some geometric hypothesis.



Title: Three circles theorem for volume of conformal metrics

Abstract: In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the Qg curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics. This is a joint work with Zihao Wang.



Title: Rigidity of minimal graphs over Euclidean half-space with constant Neumann boundary value

Abstract: We talk about the rigidity of solutions to the minimal surface equation with constant Neumann boundary value in Euclidean half-space, and we prove that these solutions are affine functions under the assumption of one-sided linear growth at infinity.



Title: Kähler-Ricci flow on minimal Kähler manifolds

Abstract: A minimal Kähler manifold is a compact Kähler manifold of nef canonical line bundle, on which the Kähler-Ricci flow starting from any initial Kähler metric admits a long-time solution. It is a fundamental problem to understand their long-time behaviors and singularities. In this talk, we shall give an overview on recent developments on this topic.



Title: Rigidity Theorem for Poincaré-Einstein Manifolds

Abstract: In this talk, I first introduce the classical rigidity theorem for Poincaré-Einstein manifold, which has conformal compactification in high regularity. Then I will report some recent rigidity result for Poincaré-Einstein manifold in the upper half plane model, which take the Euclidean space as its conformal infinity and whose adapted conformal metric has quadratic curvature decay at infinity. This is joint work with Sanghoon Lee (KIAS).



Title: Non-convexity of level sets of k-Hessian equations in convex ring

Abstract: In this talk, we will construct explicit examples that show the sublevel sets of the solution of a k-Hessian equation defined on a convex ring do not have to be convex. This is a joint work with Zhizhang Wang.