9 papers
Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
Guohuan Qiu, Jin Yan
We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). M…
Robin and Neumann problems for the graph scalar curvature equation
Guohuan Qiu
We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumptio…
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
Jiahuan Li, Xi-Nan Ma, Guohuan Qiu +1
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict -convexity of the solution for the corresponding tors…
Nonconvex Sublevel Sets for the Three-Dimensional Special Lagrangian Equation
Guohuan Qiu
For each phase , we construct a smooth, bounded, uniformly convex domain in for which the zero-Dirichlet solution of the special Lagrangian equation has…
Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
Jiahuan Li, Yilu Liu, Xi-Nan Ma +1
Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such tha…
Some counterexamples for the special lagrangian curvature equation
Guohuan Qiu, Guanyu Tao
We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant pos…