https://neckpinch.com/authors/kai-xu/New arXiv papers by Kai Xu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.06698arxiv:2607.06698Tue, 07 Jul 2026 00:00:00 GMTKai Xu — An ∞-harmonic function is a viscosity solution of ∇² u(∇u,∇u)=0, or equivalently, an absolute minimizer of \|∇u\|_(L^∞). We prove a variety of new structural and regularity results in two dimensions, including: 1. ∞-harmonic functions in domains of ℝ² are C^(1,1/3). 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating ∞-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the p→∞ limit of the classical duality between p-harmonic and q-harmonic functions in ℝ², where 1/p+1/q=1.Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesWeak & singular flowsBoundaries & non-compact https://arxiv.org/abs/2502.09727arxiv:2502.09727Thu, 13 Feb 2025 00:00:00 GMTOtis Chodosh, Yi Lai, Kai Xu — We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be ℝ³. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureWeak & singular flows https://arxiv.org/abs/2405.15181arxiv:2405.15181Fri, 24 May 2024 00:00:00 GMTKai Xu — We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish C^(1,α) regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.Mean curvature flowOther curvature flowsInverse mean curvature flowSingularitiesWeak & singular flowsStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2307.01922arxiv:2307.01922Tue, 04 Jul 2023 00:00:00 GMTKai Xu — Let M be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If M has nonvanishing second homotopy group, then it is known that the π₂-systole of M (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most 8π. We prove the following gap theorem: if M is further not a quotient of S²× S¹, then the π₂-systole of M is no greater than an improved constant c≈ 5.44π. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow.Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvature https://arxiv.org/abs/2307.00725arxiv:2307.00725Mon, 03 Jul 2023 00:00:00 GMTKai Xu — We prove a new existence theorem for proper solutions of Huisken and Ilmanen's weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities