Gauss curvature flow

In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted $\alpha$-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any $\alpha >0$ the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the $\alpha$-Gauss curvature flow ${\partial }_{t}X=-K^{\alpha }\nu$, $\alpha >0$. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If $\alpha >\frac{1}{n+2}$, we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.

In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group ${\mathrm{Nil}}_3$ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.

We classify invariant surfaces in the 3-dimensional solvable Lie group $\text{sol}$ that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields $\{F_1,F_2,F_3\}$, where $F_1$ and $F_2$ generate horizontal translations and $F_3$ generates the scaling isometry. We establish rigidity results for $F_3$-invariant surfaces, proving that specific totally geodesic vertical planes are the only $F_1$- and $F_2$-solitons. For $F_1$-invariant surfaces, we establish the main geometric properties of $F_2$- and $F_3$-solitons in both the extrinsic and intrinsic Gauss curvature.

In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.

The study of the dual curvature measures [Y. Huang, E. Lutwak, D. Yang & G. Y. Zhang, Acta. Math. 216 (2016): 325-388], which connects the cone-volume measure and Aleksandrov's integral curvature, and has created a precedent for the theoretical research of the dual Brunn-Minkowski theory. Motivated by the foregoing groundbreaking works, the present paper introduces the $p$-th dual $k$-torsional rigidity associated with a $k$-Hessian equation and establishes its Hadamard variational formula with $1\le k\le n-1$, which induces the $p$-th dual $k$-torsional measure. Further, based on the $p$-th dual $k$-torsional measure, this article, for the first time, proposes the $p$-th dual Minkowski problem of the $k$-torsional rigidity which can be equivalently converted to a nonlinear partial differential equation in smooth case: \beginalign f(x)=τ(|\nabla h|^2+h^2)^{\fracp-n2}h_Ω(x)|Du(ν^-1_Ω(x))|^k+1σ_n-k(h_ij(x)+h_Ω(x)δ_ij), \endalign where $\tau >0$ is a constant, $f$ is a positive smooth function defined on $S^{n-1}$ and ${\sigma }_{n-k}$ is the $(n-k)$-th elementary symmetric function of the principal curvature radii. We confirm the existence of smooth non-even solution to the $p$-th dual Minkowski problem of the $k$-torsional rigidity for $p<n-2$ by the method of a curvature flow which converges smoothly to the solution of equation (). Specially, a novel approach for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $\Phi ({\Omega }_t)$.

In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the $\alpha$-Gauss curvature flow with $\alpha >\frac{1}{2}$, asymptotic to a round cylinder.

A $\lambda$-translator is a surface in Euclidean space ${\mathbb{R}}^3$ whose Gauss curvature $K$ satisfies $K=⟨N,\vec{v}⟩+\lambda$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $\lambda \in \mathbb{R}$. In this paper, we classify all $\lambda$-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

We solve a general class of free boundary Monge-Ampère equations given by $$\det D^{2}u=\lambda \frac{f(-u)}{g(u^{\star })h(\nabla u)}{\chi }_{\{u<0\}}\text{ in }{\mathbb{R}}^n,\nabla u({\mathbb{R}}^n)=P$$ where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.

In this paper, we consider $\lambda$-translating solitons and $\lambda$-shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only $\lambda$-translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only $\lambda$-shrinkers with constant mean curvature. We give a classification of the $\lambda$-translating solitons and $\lambda$-shrinkers with one constant principal curvature.

In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.

P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in ${\mathbb{R}}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\le k\le n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowski problem for the $k$-torsional rigidity and confirm the existence of its smooth non-even solutions by the method of a curvature flow. Specially, a new proof method for the uniform lower bound estimation in the $C^0$ estimation for the solution to the curvature flow is presented with the help of invariant functional $\Phi ({\Omega }_t)$.

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk and Schnürer from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.

In this paper, we study $K^{\alpha }$–translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space ${\mathbb{E}}^3$. First, we investigate the condition under which two parallel surfaces can become $K^{\alpha }$–translators moving with the same speed $w$. Then, we examine $K^{\alpha }$–translators on canal surfaces and we show that if a canal surface is $K^{\alpha }$–translator, then it must be a surface of revolution in ${\mathbb{E}}^3$. We also provide examples for moving a surface of revolution under $K$–flow (Gauss curvature flow) and $K^{-1/2}$–flow (inverse Gauss curvature flow) along a direction $w=(0,0,1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^{\alpha }$–translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$, while the such rotational surfaces itself is a $K^{\alpha }$–translator with speed $w$.

This article introduces the $L_p$-Gauss dual curvature measure and proposes its related $L_p$-Gauss dual Minkowski problem as: for $p,q\in \mathbb{R}$, under what necessary and/or sufficient condition on a non-zero finite Borel measure $\mu$ on unit sphere does there exist a convex body $K$ such that $\mu$ is the $L_p$ 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, $h_k$ is the support function of convex body $K$, $\nabla h_K$ and ${\nabla }^2{h}_K$ 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 $p,q\in \mathbb{R}$ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case $p,q\in \mathbb{R}$ with $q<p$ is established.

The $p$-Laplacian evolution equation and the $\alpha$-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.

In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on.

The Minkowski problem for torsional rigidity ($2$-torsional rigidity) was firstly studied by Colesanti and Fimiani using variational method. Moreover, Hu also studied this problem by the method of curvature flows and obtained the existence of smooth even solutions. In addition, the smooth non-even solutions to the Orlicz Minkowski problem $w.r.t$ $q$-torsional rigidity were given by Zhao et al. through a Gauss curvature flow. The dual curvature measure and the dual Minkowski problem were first posed and considered by Huang, Lutwak, Yang and Zhang in. The dual Minkowski problem is a very important problem, which has greatly contributed to the development of the dual Brunn-Minkowski theory and extended the other types dual Minkowski problem. To the best of our knowledge, the dual Minkowski problem $w.r.t$ ($q$) torsional rigidity is still open because the dual ($q$) torsional measure is blank. Thus, it is a natural problem to consider the dual Minkowski problem for ($q$) torsional rigidity. In this paper, we introduce the $p$-th dual $q$-torsional measure and propose the $p$-th dual Minkowski problem for $q$-torsional rigidity with $q>1$. Then we confirm the existence of smooth even solutions for $p<n$ ($p\ne 0$) to the $p$-th dual Minkowski problem for $q$-torsional rigidity by method of a Gauss curvature flow. Specially, we also obtain the smooth non-even solutions with $p<0$ to this problem.

We study the regularity of the $p$-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of the convex hypersurface near the interface.