arXiv research

The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.

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

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…

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

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbe…

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

Study translating solitons in Minkowski space with prescribed Gauss image.

Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant princi…

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…

The paper finds convex hypersurfaces with specific curvature properties.

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.

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

We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no…

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of som…

Paper solves Minkowski problem for p-harmonic measures.

We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{…

Motivated by applications in architecture and design, we present a novel method for increasing the developability of a B-spline surface. We use the property that the Gauss image of a developable surface is 1-dimensional and can be locally well approximated by circles. This is cast into an algorithm for thinning the Gau…

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed re…

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

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

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…

The curvature of Gauss maps for flat submanifolds is studied in space forms.

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as an application of the minimal surface theory. This looks an interesting phenome…

Study of light function singularities on surfaces.

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

In this paper, our purpose is to study rigidity theorems for $λ$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $λ$-hypersurfaces, we prove that an entirely graphic $λ$-hypersurface in Euclidean space is a hyperplane.

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regul…

Study on unique generalized Gauss maps of minimal surfaces sharing hypersurfaces in projective varieties.

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

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then $M$ must be an affine hyperplane.

The paper improves defect relations for Gauss maps of minimal surfaces intersecting hypersurfaces in projective space.

In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits …

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that t…

The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1 and Theorem 3.2 not only improve the previous results, but also are optimal. I…

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore…

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transvers…

Study on minimal surfaces and their Gauss maps intersecting a specific hypersurface.

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …

In this short note we study Bernstein's type theorem of translating solitons whose images of their Gauss maps are contained in compact subsets in an open hemisphere of the standard $\mathbf{S}^n$ (see Theorem 1.1). As a special case we get a classical Bernstein's type theorem in minimal submanifolds in $\mathbf{R}^{n+1…

This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.

We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a…