https://neckpinch.com/authors/na-fu/New arXiv papers by Na Fu, from Neckpinch Tue, 03 Nov 2026 08:37:45 GMT https://neckpinch.com/papers/2412.13557/arxiv:2412.13557Wed, 18 Dec 2024 00:00:00 GMTNa Fu, Jianping Sun — This article introduces the Lₚ-Gauss dual curvature measure and proposes its related Lₚ-Gauss dual Minkowski problem as: for p,q∈ℝ, under what necessary and/or sufficient condition on a non-zero finite Borel measure μ on unit sphere does there exist a convex body K such that μ is the Lₚ Gauss dual curvature measure? If K exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: \beginalign e^{-\frac{|\nabla h_K|^2+h_K^2}2}h_K^1-p (|\nabla h_K|^2+h_K^2)^{\fracq-n2} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \endalign where f is a given positive smooth function on unit sphere, hₖ is the support function of convex body K, ∇h_K and ∇²h_K are the gradient and Hessian of h_K on unit sphere with respect to an orthonormal basis, and I is the identity matrix. We confirm the existence of solution to the new problem with p,q>0 and the existence of smooth solution to the equation (0.1) with p ,q∈ℝ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case p,q∈ℝ with q<p is established.Other curvature flowsGauss curvature flowKähler geometryStability & uniqueness