arXiv research

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.

Study on the regularity of $p$-Gauss curvature flow near flat interfaces.

Study of capillary Gauss curvature flow shrinking to a point.

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.

The paper studies how convex hypersurfaces evolve under curvature flows in space forms.

Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence…

Proves uniqueness of geometric flow in various Riemannian manifolds.

Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.

Study translating solitons in Minkowski space with prescribed Gauss image.

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0<p<1$. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

The paper classifies invariant translators for a specific curvature flow.

Study curve shortening flows on specific surfaces, proving properties and existence.

The paper constructs hypersurfaces translating under powers of Gauss curvature.

The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.

The paper classifies special solitons and shrinkers in Euclidean space.

We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…

Smooth solutions up to evolving free boundaries for degenerate equations.

Paper solves Minkowski problem for p-harmonic measures.

We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

Proves long-term smoothness of curved surfaces evolving under specific curvature rules.

We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.

Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.

Solves $L_p$-Gaussian chord Minkowski problem using Gauss curvature flow.

The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior p…

Classifies surfaces translating under specific curvature flows.

The paper studies how surfaces move by mean curvature flow and what happens at singular points.

Let $F:Σ^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $γ:(Σ^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher c…

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

We address the asymptotic behavior of the $α$-Gauss curvature flow, for $α>1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determ…

Study flows for capillary Minkowski problems in half-spaces.

Study an anisotropic capillary flow to solve capillary Orlicz-Minkowski problem.

The paper explores $K^α$-translators on parallel and canal surfaces in 3D space.

Paper solves Minkowski problem for q-torsional rigidity using curvature flow.

We prove that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under the flow by any power $α>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…

Paper solves a generalized chord Minkowski problem using Gauss curvature flows.

Researchers classify and describe $K^α$-translators in Euclidean space.

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …

Paper solves Musielak-Orlicz-Gauss image problem using parabolic flows.

In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^\infty$-topology to a smooth strictly convex soliton as $t$ approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andre…