arXiv research

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

Classification of constant curvature surfaces in Berger spheres.

We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.

The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known a…

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

Study on Gauss images of specific minimal surfaces with finite curvature.

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

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

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…

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

Planes and spheres are the only stationary surfaces with constant Gauss curvature.

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

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

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}.

The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…

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…

We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb P}^{m-1}({\mathbb C})$.

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate …

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and …

Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.

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…

The paper constructs hypersurfaces translating under powers of Gauss curvature.

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundam…

Study finite curvature solutions on surfaces with nonnegative Gauss curvature.

Study Gauss maps of surfaces in Heisenberg group using hyperbolic geometry.

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

Study of capillary Gauss curvature flow shrinking to a point.

The paper classifies invariant translators for a specific curvature flow.

The paper studies hypersurfaces with constant weighted mean curvature in Gaussian space.

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

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.

Study translating solitons in Minkowski space with prescribed Gauss image.

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.

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.

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

We investigate complete minimal hypersurfaces in the Euclidean space $$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…

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

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

The paper proves properties of strain tensors on surfaces with changing Gauss curvature.

The paper improves the Gauss curvature estimation for harmonic surfaces and verifies a modified defect relation.

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

The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.

The present paper discusses that a prescribed Gauss-Kronecker curvature problem on the product of unit spheres.

Study proves surfaces with constant curvature are simple shapes.